CHO MOMO 



RUFUS CHOATE 















to 




CHO MOMO 


THE ONE ARITHMETIC 


CONTAINS 

THE ONLY ONE POSSIBLE UNIVERSAL 
NUMERICAL LANGUAGE. 


Washington, D. C. Edition 


m -'202 






Copyright 
By Rufus Choate, 
1914. 



AUG 27 1914 

©C1.A3S0141 


INTRODUCTORY. 


Arithmetic, or Momo, is the science of numbers. 
It is the alphabet upon which all other sciences are 
constructed. As well expect to read words without 
a knowledge of letters as to think to advance in 
science without knowing the value of numbers. 

It is said of the ancient Magi that they especially 
admonished their priests never to reveal truth to the 
people. They well knew that only mysticism would 
hold the masses in subjection: revelation of truth 
equalizes all. 

The scientists of this day, descendants of the 
Magi, have hidden the sacred alphabet of numbers 
under mystifying words that darken and confuse 
the novice who dares approach the path that leads 
to the Temple of Wisdom. 

Arithmetic, or Momo, is divided into two grand 
orders, namely: the abstract, and the concrete. 
This is in accordance with the universal division of 
all creation into imponderable and ponderable ele¬ 
ments. 

The abstract refers to general application, without 
individualization, as in 1, 2 , 3, 4, 5, 6, 7, 8, 9. 

The concrete specifies an individual, as: one man, 
two pounds, three books. 

Numbers are also divided into prime units, asy 0, 
3, 2 , 3, 4, 5, 6, 7, 8, 9, which are all the existing 
primes; they are wholly different from prime num¬ 
bers, which are many; and composite units, which 


4 


are of vast numbers, depending on the various com¬ 
binations of the ten prime units. 

There are also prime numbers, and composite 
numbers. A prime number is one that cannot be 
farther divided. It is an error to say that it is 
divided by one, or itself. The one leaves it un¬ 
changed as though it had not been tried. The only 
unit, or entity, (which are identical) that can de¬ 
stroy a prime number, is itself, and there is not 
another power in the entire universe that can cause 
this annihilation. 

It is wholly wrong to call 4, 6, 8, or 9, a composite 
number under the supposition that because 4 is the 
product of 2 X 2, that while 6 is the product of 2 X 
3, 8 is that of 2 X 2 X 2, and 9 is that of 3 X 3. 
This does not make a composite. Each prime unit 
as 2, 3, 4, 5, 6, 7, 8, 9, owes its existence wholly and 
alone to the possession of the value of 1. 2, pos¬ 

sessing two ones, 3, three ones, 4, four ones, 5, five 
ones, 6, six ones, 7, seven ones, 8, eight ones, and 9, 
nine ones. 

A composite even number is one that can be 
divided by 2, or 5, or their multiples. A composite 
uneven number is one that is divided by another 
number than 2, or one of its multiples. 

A unit is one, a part of one, or a multiple of one. 

All creation is divided into two great classes, 
namely: imponderable and ponderable substances, 
and every thing must come under one of these two 
orders. 


5 


The cipher represents all imponderable matter, as 
force, energy, poverty, honor, and in illimitable 
space has no power, while its ability increases with 
geometrical ratio as it comes into contact with con¬ 
finement or resistance, or is adjoined to a unit. 
Thus standing alone, o, is, as it were, nothing, with 
a 10, it is increased ten fold, confined closer as 100, 
it is one hundred fold, 1000, one thousand times 
more powerful; with a 2, it is 20, 200, 2000; with a 

5, 50, 500, 5000. 

All ponderable qualities must have a standard of 
unit; that comparison with others can be made. It 
may be infinitely small, or infinitely large; yet large 
or small, while ponderable it is material in any and 
in all quantities. And the imponderable, just as 
substantial, are immaterial. 

Quantity is the measurement of a ponderable ele¬ 
ment. An imponderable quantity cannot be 
measured except through the quantity of resistance 
excited by the ponderable. 

The province and use of arithmetic is to measure 
the varying quantities of ponderable matter. It has 
nothing to do with the imponderable, except in so 
far as the imponderable acts upon the ponderable. 

Numbers are also known as cardinal, 1 , 2, 3, 4, 5, 

6, 7, 8, 9, and ordinal, as: 1st, 2nd, 3d, 4th, 5th, 
6th, 7th, 8th, 9th. They are whole, as: 1, 2, 3, 4, 
5, 6, 7, 8, 9. Fractional, as: %, %, %, etc., or 
mixed, as: l- 1 ^, 2-%, 3-%, etc. 


6 


CHO MOMO. 

The especial object desired to be obtained in creat¬ 
ing Cho Momo is to have a common mathematical 
language for the entire world; a language any one 
can speak, and that every one shall speak alike. 

This language is called Cho (always pronounced 
ko) and Momo is the universal arithmetic. 

Cho will steadily advance through the alphabet of 
numbers to the higher mathematics, proving itself 
thoroughly efficient for a giant’s task. 

This new language selects none of its constituents 
from any existing or dead tongue. It arbitrarily, to 
suit its own genius, advances creating a living, ever- 
existing, ever-enlarging language. 

It is evident to every student of political economy 
that a one government, some day, not far distant, 
is to reign over the people. Its coming is promised 
in the Sacred Word. Its mentalization is shaking 
thrones and creeds of the present. It will give to 
humanity the hope of far better things, not because 
of the asking, but because of the right: not as a 
favor, but as a principle of integrity; not because 
humanity is worthy, but because God is good. 

That which is to come cannot exist through force. 
Like truth itself, which cannot come into activity 
until the false has been thoroughly tried, and has 
proven its own impotency, the one future govern- 


7 

ment will exist because it will be the better way of 
managing affairs. 

Speaking axiomatically, the inevitable is irresisti¬ 
ble. Since the coming of a one government is 
inevitable, an universal language is an essential to 
its life; therefore, the present Cho system, being an 
essential, and if it is to be universal, is an equal and 
possesses the quality of the inevitable, that is to say, 
it is irresistible. 

While it is intended to translate Cho Momo into 
every known tongue, to give it a lip, and a sign lan¬ 
guage, the words in the present advance English 
edition are identical with those that will appear in 
other translations. 

Cho Momo will be introduced in two parts, each 
copy of which is separately numbered, and there is 
no duplicate of any number of an edition. If any 
copy of Cho‘Momo appears without its individual 
number, it is without legality, and a duplicate will 
bring liability to indictment. 

Any of the contents of Cho Momo can be freely 
used socially, or for scholastic instruction, or in re¬ 
views and criticisms, (which are earnestly solicited) 
but the commercial value of any publications, im¬ 
mediate or mediate, rests with the present copyright. 


8 


NOTATION. 

Notation is the art of writing numbers. 

An artist is one who does a thing in an appro¬ 
priate way. 

The right way is always the easiest way; it is the 
shortest line between two points; hence, the art of 
writing numbers, is to write them in the most ap¬ 
propriate manner. If any improvement is possible 
the prior must yield to its successor. It is a law 
of nature that nothing (mind, man, or matter) can 
cease existing until a stronger, abler, better or 
worse, successor drives it down and out. 

Also another universal law prevails with which 
every science must accord, namely: that similars as¬ 
sociate and affinities agree. 

Also, though not especially applicable to our sub¬ 
ject, another: that the environment is forever in 
the endeavor to reproduce itself. That is to say 
that every number tries to make every other number 
like itself. 

To agree with the law of association all numbers 
at all times must be arranged in accordance with 
their respective units, namely: units with units, tens 
with tens, and hundreds with hundreds. If this 
law is not observed, any method of notation starts 
with an error and must end in failure. 

This law of association comes under the mathe- 


9 


matical law that equals produce equals, that is to 
say, if the right is added to the right the result is 
favorable, but right added to wrong increases the 
wrong many fold. 

Three recognized methods of writing numbers 
at the present time exist. The Cho method, here¬ 
with introduced, will be a fourth. 

One, the Arabic method, called by the Cho, Ala, 
has ten prime units, namely: 0, 1, 2, 3, 4, 5, 6, 7, 
8, 9, and a multitude of composites resulting from 
the various combinations of these ten prime units. 

The Ala method is universally accepted. It is 
so beautifully simple and so wonderfully useful that 
no improvement can be made; hence, it is eternal. 
It is accompanied by one weakness, not in itself', but 
in its companion and associate, namely: the words 
used to express these Ala numbers are as various 
as the faces of humanity, every man speaking unlike 
the other. It is to correct this defect that the Cho 
words are created. 

The Roman method consists of seven letters used 
to express certain numbers, namely: I, for one, V, 
five, X, ten, L, fifty, C, one hundred, D, five hun¬ 
dred, and M, one thousand. 

The Roman method is restricted to the commem¬ 
oration of annual dates: the heading of chapters in 
books; in manuscript, and on monuments. Its uses 
are comparatively small and its existence is short. 

Also words are employed by each nation and 
people to express numbers. These words are as 


IO 


different as the various tongues and produce a 
babble of confusion. 

And now comes the Cho (always pronounced ko) 
method. It is to make a common mathematical 
language for all people, for all time. It is to ac¬ 
company the Ala, as a wife her husband, congenial 
and true, one to the other. 

As magnetism is the conjunction of two elements 
making them one, giving birth to life; and as elec¬ 
tricity is disjunction, separating the one into two, 
causing death: so the Cho method magnetically 
associates in conjugal bliss with its friend and com¬ 
panion, the A la, as its full equal, with equal re¬ 
sponsibilities, equal incentives and equal rewards. 

Cho is as compact as Ala, its equal in every re¬ 
spect : it is.as efficient and what one can do the other 
can accomplish. 

Following are numbers in Cho and in Ala: simple, 
rich, beautiful and true. 


THE CHO AND ALA CARDINAL NUMBERS ARE AS FOLLOWS: 


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biloe diloe filoe giloe kiloe liloe ' miloe niloe piloe bimoe 


AN E ADJOINED TO A CARDINAL MAKES IT ORDINAL. 


15 


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16 





17 


CHARACTERISTICS. 

There are many characteristics that show, how 
wonderfully the literal alphabet gives aid to the Cho 
system: as though it was formed in anticipation of 
the coming of the other. 

b and d for 1 and 2 

f “ g “ 3 “ 4 

k “ 1 “ 5 ■“ 6 

m “ n “ 7 “ 8 

p “ o “ 9 “ 10 

b, f, k, m, p, for odd numbers, and 
d, g, 1 , n, o, for even numbers. 

The a, e, i, o, u, and a, are tens with mio, nio, 
and pio. . 

a for one ten 


e 

u 

two tens 

i 

(( 

three tens 

0 

a 

four tens 

u 

(t 

five tens 

a 

u 

six tens 

mi 

a 

seven tens 

ni 

a 

eight tens 

Pi 

a 

nine tens 


B 


i8 


PARENTHESES. 

Before advancing to a higher step in our journey 
it is necessary to clear mathematics, and here, espe¬ 
cially, arithmetic, of the confusing and wholly un¬ 
explained use of the parenthesis. 

Generally the novice’s eyes are blinded by brackets 
confusing numbers in enclosures without explana¬ 
tion, and in fact without, as yet, clearly defined 
laws for the use of the several kinds of these 
brackets. 

The first parenthesis to be used for enclosing 
numbers, is the simple (), and is named bua. 

Thus: 

2 + 3X4 — 1= 13.. 

di pu fi su gi ru bi ke af. 

(2 + 3) X 4 — 1 = 19. 
di pu fi bua su gi ru bi ke ap. 

The bua asserts that the several units within it 
are to be taken as a one unit. 

In a solution this is the first parenthesis to be 
dissolved. 

The second parenthesis is [ ]. 

Thus: [10 — (2 — 3)] X I — 1, and is named 
bue. 

The third is bui, \ J thus: 
i 5 X 7 + [10 — (2 — 3)] i X 4—1. 

The fourth is buo, < > thus: 


19 


< 6 -r- 3 + \ > 5 x 7 + [ 10 — (2 — 3)] \ 

X 4 — 1. 

It is wholly undesirable at the present to go far¬ 
ther than bua ( ). When the other paientheses 

can be serviceable they will be explained. 

With regard to placing of numbers within a 
parenthesis it must be always remembered that the 
plus, or pu, sign, +, must be changed to minus, 
—, and the minus to plus, before the negative 
sign —. 

-2+3X4-1 —2—1X4-1 

— (2 — 3) X4 — 1 —(2 + 3)X4 —1 


20 


PUUM. 

ADDITION. 

The first stepping stone of arithmetic is called 
addition. 

Study it thoroughly. 

Learn all that can be learned regarding it. 

Do not consider it lightly. 

Your success in higher mathematics depends 
largely on your clear comprehension of this guide 
that stands at the door; modest, but exceedingly 
useful. 

Addition has one characteristic that distinguishes 
it from every other mathematical element. It has 
a sum. 

What is a sum? 

It is the result obtained from adding one thing 
to other things. 

The highest use of addition is to obtain this sum. 

It is necessary that one shall clearly comprehend 
the various signs used ins'tead of words to express 
mathematically the process of obtaining this sum. 

The sign + called generally plus, meaning that 
one number is added to another to obtain the sum. 
This sign is called in the Cho language, pu. 

Here, to aid in giving illustrations of this pu 
sign, another very important sign must be defined. 

It is the sign of equality, or equal to. = mean- 


I 


21 


ing that one number pu, or plus, added to another, 
equal a certain sum. The name given by Cho for 
this sign is, ke. 


I 

+ 

I 

= 

2 

144 

+ 

89 

= 

233 

bi 

pu 

bi 

ke 

di 

bi-og 

pu 

nip 

ke 

di-if 

2 

+ 

3 

= 

5 

233 

+ 

144 

— 

397 

di 

pu 

fi 

ke 

ki 

di-if 

pu 

bi-og 

ke 

fi-pim 

5 

+ 

3 

= 

8 

397 

-t- 

233 

= 

630 

ki 

pu 

fi 

ke 

ni 

fi-pim 

pu 

di-if 

ke 

li-io 

8 

+ 

5 

= 

13 

630 

+ 

397 

= 

1,027 

ni 

pu 

ki 

ke 

af 

li-io 

pu 

fi-pim 

ke 

ao-em 

13 

+ 

8 

= 

21 

1,027 

+ 

630 

= 

1,657 

af 

pu 

ni 

ke 

eb 

ao-em 

pu 

li-io 

ke 

al-um 

21 

+ 

13 

= 

34 

1,657 

+ 

1,027 

= 

2,684 

eb 

pu 

af 

ke 

ig 

al-uni 

pu 

ao-em 

ke 

el-nig 

34 

+ 

21 

— 

55 

2,684 

+ 

1,657 

= 

4,33i 

ig 

pu 

eb 

ke 

uk 

el-nig 

pu 

al-um 

ke 

of-ib 

55 

+ 

34 

= 

89 

4,331 

+ 

2,684 

= 

7,oi5 

ug 

pu 

ig 

ke 

nip 

of-ib 

pu 

el-nig 

ke 

mio-ak 

89 

+ 

55 

= 

144 

7,oi5 

+ 

4,33i 

= 

ii ,346 

nip 

pu 

uk 

ke 

bi-og 

mio-ak 

pu 

of-ib 

ke 

bi-af-ol 


11,346 + 7,015 = 18,361 
bi-af-ol pu mio-ak ke bi-nif-ab 


Addition can be made in columns when units, tens 
and hundreds, etc., are placed in their respective 
order. 


1 

bi 

2 

di 

5 

ki 

8 

ni 

1 

bi 

3 

fi 

3 

fi 

5 

ki 

2 

di 

5 

ki 

8 

ni 

13 

af 

13 

af 

21 

eb 

34 

ig 

55 

1 

8 

ni 

13 

af 

21 

eb 

34 



ig 55 uk 89 nip 


21 eb 34 



22 


89 

nip 

178 

bi-min 

422 

gi-ed 

55 

uk 

89 

nip 

928 

pi-en 

34 

ig 

55 

ig 

347 

562 

fi-om 

ki-ad 

178 

bi-min 

422 

gi-ed 



2259 

ed-up 





By far the best method for adding large and 
intricate columns of figures'that may be subjected 
to legal proceedings are by units and tens. The 
figures, orderly arranged, should be divided into 
columns of units and tens, through ruling, or men¬ 
tally. The units column should first be added and 
the tens carried over to the tens column: then the 
sum of these two should be written beneath these 
two columns. 

The same process is to be continued as though 
each were units and tens. Finally these several 
sums are to be added. 

No student should presume to continue the far¬ 
ther study of arithmetic until the method of adding 
with positive proofs of accuracy of the sum are 
thoroughly mastered. Surely no one can become 
an able mathematician without facility in adding 
with certainty of correctness. 

Often complicated mathematical problems will 
be vitiated by a slight mistake made in adding and 
occasionally the mistake made here will be equal 
to criminality. Therefore it is earnestly urged 
that each student shall thoroughly master the follow¬ 
ing and other examples. 






23 


Occasionally it is necessary that one shall be 
positive as to the accuracy of an addition. 

In mathematics nothing can be properly left to 
supposition: each statement must bear positive proof 
of accuracy. 

There are two methods only that should be legally 
accepted as to the proven correctness of the sums 
of figures; and in the handling of large public 
finances one of these two should always be made, 
namely: that of casting out the nines, or that of 
casting out the elevens. Since the nine method 
has a slight advantage in simplicity over the eleven 
only the nines need be given. 


7th 

6th 

5th 

4th 

3d 

2nd 

1st 

9 

87 

46 

25 

32 

64 

10 




64 

25 

39 

02 



25 

32 

47 

63 

19 


1 

23 

45 

67 

89 

10 






12 

21 





19 

14 

31 




25 

62 

45 

29 



9 

31 

47 

36 

28 


4 

56 

78 

25 

31 

02 

29 

34 

55 

67 

82 

91 

50 

1 

02 

07 

09 

84 

73 

21 







12 






J 

45 





6 

27 

82 




7 

37 

72 

28 


24 


3 

57 


90 


6 

5 31 

3 93 

2 71 

1 28 

39 

40 30 74 98 37 60 +90 

CASTING OUT TH£ NIN£S. 

Since simplicity is the highest achievement of 
ability the casting out nines should have the most 
careful attention of the student. 

No figures of whole numbers should be admitted 
before courts of justice when large amounts are in¬ 
volved and where accuracy is a necessity without 
having been proven correct by the test of nines. 

The casting out nines is applicable to any com¬ 
bination of whole numbers, addition, subtraction, 
multiplication or division. 

Here we are considering only addition. 


1 2345 
6 7 8 9 0 
9 8 7 6 5 
432 1 0 


2 + 1 = 3 
210 =3 


2 + 2 = 4 
220 =4 


2 1 0 


220 


2 = 2 
20 = 2 


20 


222210 


3 + 4 + 2 = 9 
22 22 1 0 = 9 








25 


6th 5th 4th 3d 2d 1st 

9 87 62 34 47 29 

67 82 95 14 21 

3 45 28 76 

1 23 45 67 

29 

1 35 

2 45 26 

28 34 47 

8 76 52 32 

98 87 76 65 43 

10 12 14 16 17 18 

1 23 45 67 89 10 

64 87 92 23 

4 56 62 26 31 

12 

1 25 

45 19 21 

76 25 92 03 

1st 

2 + 6 = 8 

548 = 8 

2nd 

5 + 5 = 1 

617 =1 

3d 

7 + 8 = 6 

681 —6 

4th 

0 + 3 = 3 

498 =3 

5th 

8 + 4 = 3 

291 =3 

6th 

1 + 1 = 2 

20 =2 

5 48 

6 86 67 

4 98 1 

2 91 

20 

8 + 1 + 6 + 3 + 3 + 2 = 5 


22 96 04 87 72 48 = 5 




26 


RUUM. 

SUBTRACTION. 

The second guard at the outer door of the Temple 
of Wisdom, the companion of addition, is subtrac¬ 
tion. 

The one examines the novitiate as to his positive 
qualities, this, as to his negative principles: the one 
as important as the other. 

The one adds, this subtracts. 

He who wisely enters will carefully study these 
two guards and gain their permission to advance 
deeper into the wonderful science of numbers. 

While addition has one characteristic, namely: 
the sum; subtraction has three inseparable qualities. 
It must have a minuend or a number that is to be 
lessened, or diminished, which Cho calls, rui; the 
number that lessens or diminishes, the subtrahend, 
or, called ru, and a difference or remainder, which 
is called ruo. 

Thus: 

1 2 3 4 5 minuend, or rui 
1 2 3 4 subtrahend, or ru 

11111 the difference, or ruo. 

The one characteristic sign of addition is +, 
plus, or pu; and the one characteristic sign of sub¬ 
traction is — minus, or ru. 



27 


These two signs are so simple, so familiar and 
so useful that as no improvement therein is pos¬ 
sible, they remain eternal. 

Occasionally and especially in working through 
higher mathematics a mistake made in subtraction 
is as injurious as a mistake made in addition would 
be. Also in making laws for appropriation of public 
moneys care must be taken as to the absolute cor¬ 
rectness of these two guards, addition and sub¬ 
traction. 

Legally the only proof of correctness should be 
the test of the wonderful nine. 

. If the nines of the subtrahend plus the nines of 
the difference are equal to the nines of the minuend 
all the three characteristics are correct. 

Thus: 

1 2 3 45 = 6 
1 2 34 = 1 

1 1 1 11 = 5 + 1 = 6 

This method of casting out the nines is much 
quicker, easier and more exact than the old obsolete 
direction for making a proof. 

Each pupil is earnestly requested to practice by 
examples made by himself, under the guidance of 
this guard at the outer door before going farther. 



28 


SUUM. 

MULTIPLICATION. 

At the inner door of the Temple are two other 
guards: the name of one is multiplication, suum. 

Suum has three characteristics, namely: the mul¬ 
tiplicand or sui, the multiplier, or su, and the prod¬ 
uct, or suo. 

Sui and su are said to be factors of the product, 
suo, because they are considered as component parts 
of the product. The essential constituents of the 
sum, (puo) ; in addition, (puum) ; those of sub¬ 
traction, (suum); and those of division, (tuum) 
are not called factors. 

Factors are said to be the multipliers that pro¬ 
duce a product. 

The sign of multiplication, thus: X is called su, 
and tells that the multiplicant is multiplied by the 
multiplier. 

The product is of a like unit with its multiplican 
and the multiplier must always be an abstract. 

It is not proper to say five men times ten; but it 


29 


is, ten times five men: not ten multiplied by five 
men; but five men multiplied by ten. It is known 
that the ten refers to men, but the utterance of that 
fact is not correct. 

It is a mistake to multiply by the powers of a 
factor of any number. It only adds confusion. 
8 is one thing, and 2 3 is wholly a different mathe¬ 
matical element. 


bi 

di 

fi 

gi 

ke 

li 

mi 

ni 

Pi 

ao 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

di 

gi 

li 

ni 

ao 

ad 

ag 

al 

an 

eo 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

fi 

li 

pi 

ad 

ak 

an 

eb 

eg 

em 

io 

3 

6 

9 

12 

15 

18 

20 

24 

27 

30 

gi 

ni 

ad 

al 

eo 

eg 

en 

id 

il 

do 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

ki 

ao 

ak 

eo 

ek 

io 

ik 

do 

ok 

uo 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

li 

ad 

an 

eg 

io 

il 

od 

on 

ul 

ao 

6 

12 

18 

24 

30 

36 

42 

48 

56 

60 

mi 

ag 

eb 

en 

ik 

od 

op 

ul 

ag 

mio 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

ni 

al 

eg 

id 

do 

on 

ul 

ag 

mid 

nio 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

pi 

an 

em 

il 

ok 

ug 

ai 

mid 

nib 

pio 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

ao 

eo 

io 

do 

uo 

ao 

mio 

nio 

pio 

biko 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 


30 


7 8 6 5 7 3 4 = 4 
1 2 345 6 7 = 1 


55060138 

47194404 

39338670 

31462936 

23597202 

15731468 

7865734 


78 
17 1 
3 6 1 
373 
204 
16 9 


9 7 1 0 7 7 6 6 2 7 1 7 8 = 5 

Every product of importance, that is to say, of 
numbers liable to appear in a court of justice should 
be subjected to the proof of the nines. 

One peculiarity of the nines, never before the 
present especially pointed out, is the very evident 
fact that proof by nines when either factor is less 
than nine is unserviceable. Thus: 6 X 6 = 36, 
li su li ke il. The nines of the factors equal 3: 





3i 


while those of the product equal 0. When each of 
the factors is more than nine we get the proper re¬ 
sult from the proof. 

Thus: 


3 6 = 9 = 0 
3 6 = 9 = 0 


2 1 6 = 9 = 0 
108 =9 = 0 


1296 = 9 = 0 
1 2 9 6 = 9 = 0 


7 7 7 6 = 9 = 0 

1 1 6 6 4 =9 = 0 

2 5 9 2 =9 = 0 

1 2 9 6 =9 = 0 


1679616 = 9 = 0 
1679616 


10065696 

1679616 

10065696 

15115544 

11757312 

10065696 

1679616 


272040769545 6 = 3 








32 


Here an error exists. Where is it? 

The author of Cho Momo does not deem it nec¬ 
essary to fill space (a defect of so many arithmetics) 
with silly, useless additions, subtractions, multipli¬ 
cations and divisions that any student can readily 
make at any time; but it is earnestly requested that 
every seeker of knowledge and ability will drill 
thoroughly in the art of the four fundamentals of 
mathematics. 

CONTRACTIONS OF MULTIPLICATION. 

It must be remembered that the shortest road is 
not always the safest road. 

Mathematics have many more higher functions 
than is usually supposed. They are not given simply 
for attaining an end easily, but are given as the one 
true method of producing firmness, stability, health 
and purity in the bony structure of the human system. 

The athlete may train as he is ordered, diet ac¬ 
cording to rule, but most surely if he has not a 
knowledge of mathematics in his head his bones will 
not endure severe trial. A gentleman of culture has 
the advantage over brute strength every time. Bread 
is good, but we live not by bread alone. He who 
seeks the shortest cut across is often the last one 
home. 

Contractions in multiplications will be learned as 
the exigencies of the case demand: They are not 
worthy of especial study at this time. 


33 


First learn to multiply accurately, learn patiently, 
learn thoroughly. Every hard effort made now 
clears future problems immeasurably. 

Most of the arithmetics at present existing give 
in the study of multiplication powers and exponents 
of numbers as multipliers. This is another error 
of the many thousands committed. Powers and ex¬ 
ponents belong to the properties of numbers. To 
ask the student to jump from the ground to the 
pinnacle of the Temple without a ladder is to ask 
an impossibility. 


c 


34 


TUUM. 


DIVISION. 

Division like a beautiful goddess worthy of ado¬ 
ration, this companion of multiplication stands 
firmly established at the inner door of the Temple 
inviting approach. 

Addition, subtraction and multiplication have al¬ 
lowed us to pass this far. Will she smilingly give 
permission to enter farther? 

Cho’s name for division is tuum. Thus we have 
now, addition or puum, subtraction, ruum, multi¬ 
plication or suum and division or tuum. 

Tuum has four characteristics. First the divi¬ 
dend or the unit that is to be divided, called by Cho, 
tui; second, the divisor or tu; third, the quotient 
or tuo and tuai the remainder, in case such exists. 

A certain restriction pertains to its acts, namely: 
that only similars or like units can be divided by 
similars or likes. 

The sign of division called tu is simplicity 
itself. 

As a matter of convenience division is divided into 
short and long division. 

Short division applies to all numbers where the 
divisor is one of the prime units, namely: 2, 3, 4, 
5, 6, 7, 8, 9. 

It is an error to say that a unit is divided by 1 or 
itself. 1 or unity does not divide anything. It, 
like Deity, does not have to destroy, nor change 


35 


itself or another. As it was in the beginning per¬ 
fect, so at the end, and so also along all the path 
between the beginning and the end perfection rules 
unchanged. 

A unit does not divide itself. Its action on itself 
proves mathematically that the only power in the 
universe capable of annihilating is one’s self. One 
or Deity will not do it, another unit divides, or 
multiplies its powers; but itself alone can destroy. 

6645)8 98792033542835 (1323105123 
6 64 5 


21470 
1 9 9 3 5 


1 5 3 5 3 
1 3 2 9 0 

2 0 6 3 3 
1 9 9 3 5 


6 9 8 5 
6 6 4'5 


34042 
3 3 2 2 5 


8173 
6 6 4 5 


1 5 2 8 3 
1 3 2 9 0 


1 9 9 3 5 
1 9 9 3 5 









36 


The constituents that makes the product in multi¬ 
plication are called factors, or sua. Those that give 
the sum in addition are magnets or pua, and the 
dividend and the divisor that form the quotient are 
electrics or tua. The minuend and subtrahend are 
non-magnets rua. 

It will be observed that division engages multi¬ 
plication and subtraction in every step of its opera¬ 
tion. 

The electric 6645, in the above example, becomes, 
through the act of the factor 1, a factor in the prod¬ 
uct 6645; then it becomes a non-magnet subtracting 
its power from the main electric. 

Again the electric 6645 becomes, through the act 
of the factor 3, a factor in the product 21470, when 
it forms a non-magnet, subtracting the power of 
21470 from the main electric. 

Finally, 6645 and 1323105123 are factors capable 
of producing 8792033542335. 

In division one very important fact, not previ¬ 
ously observed by any mathematician, is that when 
a trial number is placed in the quotient for multiply¬ 
ing the divisor to obtain the product that is to be 
subtracted from the dividend it is not the trial num¬ 
ber in the quotient that should be used to multiply 
the divisor but it is the given divisor that should 
multiply the trial number. This may seem a trifling 
matter to inexperienced mathematicians. It is actu¬ 
ally an important essential in many intricate prob¬ 
lems. It is immaterial in most problems as to 


37 


whether the divisor is the multiplican or the multi¬ 
plier, but it is of serious import in higher mathe¬ 
matics. 

Ten or any multiple of ten makes the units figure 
of its main electric a portion of ten, thus: 1234 
divided by a sub-electric 10 becomes 123 X 4 X prefer¬ 
ably expressed by a period, which will be explained 
later, thus: 123.4 

100 makes the tens and units figures of the main 
electric a portion of 100, thus: 12.34 

Then it is evident that 10, or a multiple of 10, 
divides off the main electric as many figures as there 
are ciphers in the sub-electric. 

An equivalent, ike, is the xx , the xx -g-, or the ytnro 
of any number. 

1 is the ike of 10; 100, 1000, etc. 

2 ” ” ” ” 20 , 200 , 2000 , ” 

3 ” ” ” ” 30, 300, 3000, ” 

4 ” ” ” ” 40, 400, 4000, ” 

5 ” ” ” ” 50, 500, 5000, ” 

67 ” ” ” ” 670, 6700, ” 

89 ” ” ” ” 890, 8900,. ” 


38 


DIVISORS. 

The Common Divisor. 

The Greatest Divisor. 

The Greatest Common Divisor. 

All the possible factors of any number are di¬ 
visors of that number. 

Prime numbers are not made of factors and have 
no divisors. When arithmeticians say that a prime 
can be divided by 1 or itself we hold they err. One 
and the number itself are not included as divisors 
of a given number. 

A common divisor of‘two or more numbers is the 
number that will divide each. 

The greatest divisor is one that will divide each 
and be the highest divisor of the lowest of the given 
numbers. 

The greatest common divisor (g. c. d.) is the 
highest divisor common to all the given numbers. 

Prime numbers have no divisors. 

The first product of primes have only one divisor, 
as 25 has only 5. 49, only 7. 121, has only 11. 

The second product of primes has but two di¬ 
visors, as: 14641 has only 121 and 11. 

The product of primes and composites has as 
many divisors as there are multiples of the primes 
plus the several factors of the composites. 


39 


The highest divisor of any number is its one half. 

The second highest its one third. 

” third ” ” ” fourth. 

” fourth ” ” ” fifth, etc. 

Many problems of higher mathematics depend on 
the accuracy of divisors and especially of the g. c. d. 
It is the province of arithmetic to provide methods 
for obtaining these several divisors. 

It is profitable to examine the divisors of all 
numbers from 1 to 100, since all other numbers are 
multiples of these. 

From 1 to 100 inclusive the following are primes, 


1 

11 

31 

41 


61 

71 


3 

13 


43 

53 


73 

83 

5 








7 

17 

37 

47 


67 


97 


29 



59 


79 

89 


and hence have no divisor. 

The following formed by the multiple of a prime 
by itself has but one divisor. 

2 has 1, 9 has 3, 25 has 5, 49, 7, 81, 9. 

It is observed that all primes end in 1, 3, 7 or 9 
and that prime numbers of whatever magnitude, 5 
excepted, end in 1, 3, 7, or 9. 

To illustrate: 


1 

3 

5 

7 

19 

11 

13 


17 

29 

31 

43 


37 

59 

41 

53 


47 

79 

61 

78 


67 

89 

71 

83 


97 



4 o 


The common divisor of two or more numbers is 
any number that will divide each of the given num¬ 
bers. 

The greatest common divisor is the largest di¬ 
visor of the smallest of any two or more given 
numbers that will divide the other given numbers. 
In case the smallest number is itself a divisor of the 
other number it is the g. c. d. 

Thus: 8)8, 32, 64 
4 8 

8 is the g. c. d. 

In 8, 12, 16; since 8 will not divide 12, one half 
of 8 or 4 is the g. c. d. 

4)8, 12, 16 
2 3 4 

It follows then that the highest fractional part of 
the lowest given number, if this number is not itself 
the common divisor, is the g. c. d. of all the given 
numbers. 

Occasionally it occurs that the smallest of the 
given numbers is a prime number having no frac¬ 
tional parts. If this prime cannot be used as the 
common divisor of the other numbers, there is no 
common divisor of those numbers. 

Occasionally one of the larger numbers is a prime 
in which case there is no common divisor. 

We are now prepared to make acquisitions. 

Common divisors of a number are necessary in 




41 


the ordinary affairs of life and in the solution of 
many mathematical problems. 

Remember that always all the possible factors of 
any number are divisors of that number. 

The first of the whole numbers that has factors 
and divisors is 2. 

1X1=2 
2 -5-1 = 1 
Then 4 

2X2 = 4 
1 = 2 = 2 

6 

2X3 = 6 
6 -f- 2 = 3 

8 

2 X 4 = 8 

8 -f- 2 = 4 
9 

3 X 3 = 9 

9 ^-3 = 3 

10 

5 X 2 = 10 
10-5-2= 5 

With the number 12 we enter an increase of fac¬ 
tors and divisors. 12 has more than 2 and 6, hence 
we have to look deeper. 

The greatest divisor of any number is its one half, 
2)12, 6, 3 


3) 4 2 



42 


6 as the greatest divisor of 12, and 6, 4, 3, 2 are all 
its possible divisors. 

When expression is given to all the divisors of 
any number the expression should be with decreas¬ 
ing ratio, thus: 6, 4, 3, 2 rather than with increasing 
ratio, as: 2, 3, 4, 6. 

To know the different divisors of a number, the 
common divisors of several numbers and especially 
the greatest common divisor is a matter of extreme 
importance in many problems. 

What are the divisors of 24? 

2) 24, 12, 6, 3 

3) 8, 4 

4) 6, 3 

Ans. 12, 8, 6, 4, 3, 2 

What of 68 ? 

2)68, 39 

Ans. 39, 2 

What of 64? 

2)64, 32, 16, 8, 4, 2 

4) 16, 8, 4, 2 

8) 8, 4 

Ans. 32, 16, 8, 4, 2 = 5 

Of 60? 


2)60, 30 

, 15 

3) 20, 

10 

4)15 


5) 12, 

10, 3 

6)10, 

5 


Ans. 30, 20, 15, 12, 10, 5, 4, 3, 2 = 9 

Now compare the above clear, concise and simple 
method with the following long and complicated 
way advised by so-called standard arithmetics. 


43 


“ The Greatest Common Divisor or Measure. 


FIRST OPERATION. 

24==2X2X2X 3 

88 = 2X2X2X11. 
2 X 2 >; 2 = 8. Ans. 


“ The greatest common divisor or measure of two or more numbers 
is the greatest number that will divide each of them without a remainder. 
Thus, 4 is the greatest common divisor of 8, 12, and 16. 

To find the greatest common divisor of two or more numbers. 

Ex. 1. Required the greatest common divisor or measure of 24 
and 88. Ans. 8. 

Resolving the numbers into their prime 
factors, thus, 24 = 2X2X2X3, and 
88 = 2X2X2X 11 = 88, we find the 
factors 2X2X2 are common to both. 
Since only these common factors, or the 
product of two or more of such factors, 
will exactly divide both numbers, it follows that the product of all 
their common prime factors must be the greatest factor that zuill exactly 
divide both of them. Therefore, 2 X 2 X 2 = 8, the greatest com¬ 
mon divisor required. 

The same result may be obtained by a sort of trial process, as by 
the second operation. 

It is evident, since 24 cannot be exactly 
divided by a number greater than itself, if it 
will also exactly divide 88, it will be the 
greatest common divisor sought. But, on trial, 
we find 24 will not exactly divide 88, there 
being a remainder, 16. Therefore 24 is not 
a common divisor of the two numbers. 

We know that a common divisor of 16 and 
24 will, also, be a common divisor of 88 
(Art. 197). We next try to find that divisor. 

- It cannot be greater than 16. But 16 will not 

exactly divide 24 there being a remainder 8; 
therefore 16 is not the greatest common divisor. 

As before, the common divisor of 8 and 16 will be the common 

divisor of 24 and 88 (Art. 197); we make trial to find that divisor, 

knowing that it cannot be greater than 8, and find 8 will exactly 

divide 16. Therefore 8 is the greatest common divisor required. 

The last method may be often contracted, if 
there should be observed to be any prime factor 

in a remainder which is not common to the pre¬ 
ceding divisor, by canceling said factor. Thus, in 
the third operation, the factor 2 being found in 
the remainder 16 once more than in the divisor 24, 
we cancel one 2 from 16, and, having left the 
composite factor 8, we divide 24 by that factor. 
There being no remainder, 8 is the greatest com¬ 
mon divisor, as before obtained.” 


SECOND OPERATION 

24 ) 88(3 
7 2 

16 ) 24(1 
1 6 

8)16(2 
1 6 


third operation. 
2 4) 8 8 (3 
7 2 

n 

8)24(3 
2 4 


44 





In consideration of the inevitable fact that within 
sixty days after the publication of Cho Momo every 
other arithmetic will be useless and so much waste 
paper the following is copied, by word and by letter, 
to show how not to do it. 


“ LEAST COMMON MULTIPLE. 


“A common mutiple of two or more numbers is a number that can 
be divided by each of them without a remainder; thus, 14 is a com¬ 
mon multiple of 2 and 7. » 

The least common multiple of two or more numbers is the least 
number that, can be divided by each of them without a remainder; 
thus, 12 is the least common multiple of 4 and 6. 

“A multiple of a number contains all the prime factors of that 
number; the common multiple of two or more numbers contains all 
the prime factors of each of the numbers; and the least common mul¬ 
tiple of two or more numbers contains only each prime factor taken the 
greatest number of times it is found in any of the several numbers. 
Hence, 

1. The least common multiple of two or more numbers must be the 
least number that will contain all the prime factors of them, and none 
others. 

2. The least common multiple of two or more numbers, which are 
prime to each other, must equal their product. 

3. The least common multiple of two or more numbers must equal 
the product of their greatest common divisor, by the factors of each 
number not common to, all the numbers. 

4. The least common multiple of two or more numbers, divided by 
any one of them, • must equal the product of those factors of the 
others not common to the divisor. 

“ To find the least common multiple of two or more numbers. 

Ex. 1. What is the least common multiple of 8, 16, 24, 32, 44? 

. Ans. 1056. 

Resolving the numbers 
into their prime factors, 
we find their different 
prime factors to be 2, 3, 
and 11. The greatest 
number of times the 2 
occurs as a factor in any 
1056 Ans. of the given numbers is 
5 times; the greatest 
number of times 3 occurs in any of the numbers is once; and the 
greatest number of times the 11 occurs in any of the numbers is once. 
Hence 2, 2, 2, 2, 2, 3, and 11 must be all the prime factors necessary 


first operation. 

8 = 2 X 2 X 2 
16 = 2X2 X 2X2 
2 4 = 2 X 2 X 2X3 
32 = 2X2 X 2X2X2 
44 = 2X2X11 
2X2X2X2X2X3X11 




45 


2 ) 8 

1 6 

2 4 

3 2 

4 4 

2 ) 4 

8 

1 2 

1 6 

2 2 

2 ) 2 

4 

6 

8 

1 1 

2 ) 1 

2 

3 

4 

1 1 

1 

1 

3 

o 

1 1 


2X2X2X2X2X3XH = 1056 Ans. 


in composing 8, 16, 24, 32, and 44; and consequently, 1056, the product 
of these factors, is the least common multiple required (Art. 202). 

Having arranged the numbers 
second operation. on a horizontal line, we divide 

by 2, a prime number that will 
divide two or more of them with¬ 
out a remainder, and write the 
quotients in a line below; and we 
continue to divide by a prime 
number as before, till the di¬ 
visor and remainders are all 
prime to each other. Then, these, 
since they include all the fac¬ 
tors necessary to form the given 
numbers and no others, we mul¬ 
tiply together for the required least common multiple, and obtain 1056, 
as before. 

The least common multiple of two or more numbers may be found 
generally by a process much shorter than either of the above methods 
by canceling any number that is a factor of any other of the given 
numbers, and also by dividing the numbers by such a composite number, 
as may be observed to be their common or greatest common divisor. 

Thus, in the third operation, 8 be¬ 
ing a factor of several of the num¬ 

bers, and 16 being a factor of one 
other number, we cancel them; and 
observing that 4 is the greatest com¬ 
mon divisor of the remaining num¬ 

bers, we divide them by it. We next 
divide by 2, as in the second operation. 

The numbers in the lower line then 
being prime to each other, we multiply them and the divisors together, 
and obtain 1056 as the least common multiple. 

The fourth operation exhibits a process 
yet more contracted. The 8 and 16 being 
factors each of one or more of the other 

numbers, we cancel them, as in the third 

operation. Of the remaining numbers we 
cut off 24 by a short vertical line from 
the rest as a factor of the least common 
multiple sought. We then strike out of the two remaining numbers 
the largest factor each has in common with the 24. by dividing each 
of them by the greatest common divisor between it and 24, and write 
the result beneath. The numbers in the lower line having no factor 
in common, we carry the process no further. The continued product 
of the number cut off by the numbers in the lower line gives 1056, the 
least common multiple, as by the other methods. In this instance we 
cut off the 24, but either the 32 could have been separated from the 
rest, or the 44 cut off, and the needless factors striken out with like 


THIRD OPERATION. 

4)0 X0 24 32 44 


2 ) 6 


8 11 


4 11 


4X2X3X4X11 = 1056 Ans. 


FOURTH OPERATION. 

0 XP 24 I 3 2 44 


24X4X11 


4 

1056 


1 1 
Ans. 









46 


result. If, however, we had cut off the 44, the numbers placed in the 
second line would have contained factors common to each other, so 
that it would have been necessary in that line to have cut off and 
stricken out factors as before. The reason for this abridged process 
is, that by the separating off, and by the striking out of factors, we 
get rid, in an expeditious way, of the factors not required to form the 
least common multiple sought. 

“Rule 1 .—Resolve the given numbers into their prime factors. The 
product of these factors, taking each factor only the greatest number of 
times it occurs in any of the numbers, will be the least common multiple. 
Or, 

“ Rule 2. —Having arranged the numbers on a horizontal line, can¬ 
cel such of them as are factors of any of the others, and separate some 
convenient one from the rest. Reject from each of the numbers remain¬ 
ing the greatest factor common to it and that number, and write the 
result in a line below. Should there be in the second line numbers hav¬ 
ing factors in common, proceed as before; and so continue until the 
numbers written below are prime to each other. The continued product 
of the number or numbers separated from the others with those in the 
last dine will be the least common multiple. 

“ Note 1.—Some give a preference to the following rule for finding 
the least common multiple: Having arranged the numbers on a hori¬ 
zontal line, divide by such a prime number as will exactly divide two 
or more of them, and write the quotients and undivided numbers in a 
line beneath. So continue to divide until the quotients shall be prime 
to each other. Then the product of the divisors and the numbers of 
the last line will be the least common multiple. 

“ Note 2.—The least common multitple of two or more numbers that 
are prime to each other is found by multiplying them together (Art. 
202 ). 

“ Note 3. —When a single number alone is prime to all the rest, it 
may be separated off, and used only as a factor of the least common 
multiple sought. 

“ Note 4.—When the least common multiple of several numbers, and 
all the numbers except one, which is prime to the others, are given, 
to find the unknown number, divide the least common multiple given 
by that of the known numbers (Art. 202).” 


After showing how not to find common divisors 
and the greatest common divisor, we present the one 
right way. 

What are the common divisors of 45 and 135? 
Take 45 as the smaller of the two numbers. 

Its lowest is 3, and 3 divides 45 into 15, its high- 


47 


est divisor; hence all the common divisors lie be¬ 
tween 3 and 15. 

3)1 3 5 
5) 45 
9 

show that all the common divisors possible to 45 and 
135 are 9, 5, 3. 

What are the common divisors of 51, 153, 255? 
Since 51 is a common divisor of 153 and 255 it is 
the g. c. d. 

The least number that will divide 51 is 3, and 3 
divides 153, 51, and 255. The only divisors of 51 
are 17 and 3 and the only common divisors of 51, 
153, 255 are 51, 17, 3. 

What are the common divisors of 180 and 360? 
180 exactly divides 360, hence it is the g. c. d. 
180 is divided by 2, hence 2 is the least common 
divisor. All the divisors common to 180 and 360 
must lie within 180 and 2, 


2) 

18 0, 

9 0, 4 5 

3 ) 

6 0, 

3 0, 15 

4) 

4 5 


5 ) 

3 6 


6) 

3 0, 

1 5 

9) 

2 0, 

10, 5 

10) 

18, 

9 

12) 

15, 



hence all the common divisors of 180 and 360 are 
180, 90, 60, 45, 36, 30, 20, 18, 15, 12, 10, 9, 6, 5, 
4, 3, 2. 




4 8 

How many common divisors have 2025, 6075, 
8100? 

Since the other numbers are divided by 2025 this 
is the g. c. d. The lowest number that divides 2025 
is 3, hence this is the least common divisor. (1. c. 
d.) All the divisors common to these three num¬ 
bers must lie within 2025 and 3. 

3)2 0 2 5, 6 7 5, 2 2 5, 7 5, 2 5 

5) 4 0 5, 1 3 5, 4 5, 1 5, 5 

5) 8 1, 2 7, 9, 3 

9) 9, 3 

2025, 675, 405, 225, 135, 81, 75, 45, 27, 25, 15, 9, 
5, 3 = 14. 

How many common divisors have 4500 and 9000 ? 

Since 4500 is an exact divisor of 9000 it is the 
g. c. d., and since it is divided by 2, this is the 1. c. d., 
hence all the divisors common to 4500 and 9000 
must fall within 4500 and 2. 


2 

3 

4 

5 


6 

9 

4500 

1500 

. . 

900 


750 

500 

2250 


. . 

450 


375 

250 

1125 

(3) 

(5) 


(9) 


125 


300 

180 


20 




150 

90 


10 




100 

60 


5 




75 

45 


4 




50 

36 


2 




25 

18 






9 


49 


4500 

300 

60 

12 

2250 

250 

50 

10 

1500 

225 

45 

9 

1125 

180 

36 

6 

900 

150 

30 

5 

750 

125 

25 

4 

500 

100 

20 

3 

450 

90 

18 

2 

375 

75 

15 



D 


35 


50 


multiples. 

The: Least Mui/tipte. 

The Least Common Mui/tipeE. 

How many people in this enlightened age know 
what a multiple is? 

How many know what a least multiple is? 

Do you? 

No. 

How many know what a least common multiple 
is? 

No one! 

Yet arithmetics are full of it. 

A multiple is the product produced by multiply¬ 
ing two or more factors. 

Thus: 3 X 5 give the product, or multiple, 15. 
2X3X4X5X6X7X8X9 produce 
the multiple, or product, 362880. There is nothing 
great nor less about the 15, or the 362880 and 
surely there is nothing common in them. They are 
the only product those numbers are capable of pro¬ 
ducing. 

In arithmetic and in higher mathematics there is 
frequently need of a number that several given 
numbers can exactly divide. A certain way has 


5i 


been discovered (how, when or through whom no 
one knows) that readily supplies this need. 

Fifteen is the only result possible from multiply¬ 
ing 3X5, and 362880 is the only number that can 
be produced by multiplying the several above given 
numbers. It is a law that the element that is es¬ 
sential in forming any objective can by its removal 
dissolve the thing previously formed. Hence the 
number 362880 can be divided and destroyed by 
any or all, of the above named factors used as di¬ 
visors. 

The requirement is not a multiple, as is 362880, 
or 15 but the only one number, regardless whether 
it is lesser or greater, that the above different num¬ 
bers can exactly divide. There is no other than a 
certain one number in existence; then why qualify 
it by the adjective, “least.” 

What is the only one right way, previously dis¬ 
covered, for finding this one much desired number? 

A law, which Cho Momo takes the credit for ex¬ 
pressing in words, is that: the product of the di¬ 
visors and the primes of several given numbers is 
the one number that each of the given numbers will 
exactly divide. 

Thus, in 3)1 5 the divisor is 3, the prime is 5. 

5 

The product of these is 15, hence 15 is the only one 
number 3 and 5 will exactly divide. 



52 


Take 

2)2, 3, 4, 5, 6, 7, 8, 9 


3) 3253749 


2) 2 5 7 4 3 


5 7 2 3 

2, 3 and 2 are divisors of the given numbers and 
since 5, 7, 2, 3 cannot be farther reduced they are 
the primes. 

2X3X2X5X7X2X3 = 2520. Hence 
2520 is the only one number the given numbers will 
exactly divide 


2520 divided by 2 


” ” ” 3 

n }} n ^ 

” ” ” 5 

” ” ” 0 

11 11 11 |y 

)> 11 11 g 

11 11 11 Q 


1260 

840 

630 

504 

420 

360 

315 

280 


One high authority has the following nonsense. 
“ The multiples that two or more numbers have 
in common are called Common Multiples, and the 
least of these, is called their Least Common Mul¬ 
tiple, (sic) which is indicated by L. C. M.” (sic) 
Is the blunder of the words driven any deeper by 
capitalizing the words “ least common multiple?” 





53 


Two or more numbers never have multiples 
(plural). They do have a multiple or product, 
which is not common but is fixed and unchangeable. 
In 3 X 5 the 15 are not multiples but is a multiple 
or product. 

Can any pupil understand the ambiguous and un¬ 
grammatical language of the above quoted author¬ 
ity? 

362880 is the multiple of the given numbers and 
2520 is the product of wholly different numbers: 
the one having' nothing in common with the other. 


54 


PROPERTIES OR CHARACTERISTICS OF 
NUMBERS. 


The prime units as previously stated are ten. 
namely: O, 0; 1, bi; 2, di; 3, fi; 4, gi; 5. ki; 6, li: 
7, mi; 8, ni and 9, pi. 

These ten are called digits after the number of 
fingers on the two hands, whence enumeration came. 

And let it be written here that originally humanity 
made calculations unerringly and unconsciously by 
instinct, as birds build their nest and bees their hive, 
wisely and well. It was only after expulsion from 
the garden of Eden that stupidity, through Adam’s 
attempt to run an independent government without 
Divine guidance, began and shows its culmination 
in the horrible production of the twentieth century 
arithmetic, which today, like all else, is passing 
through a final judgment. 

The digits began originally with the thumb of 
the left hand, which is O: the left index finger is 
1: the long finger, 2: the ring finger is 3: the left 
little finger is 4: and the little finger of the right 
hand is 5: its companion is 6: the right long finger 
is 7: the right index is 8 and the right thumb is 9. 

A units digit is the single figure standing in the 
units place, thus, 9: the tens digit is the one, as 8. 
standing at the tens place, thus, 89: the hundreds 


55 


digit is the one standing at the hundreds place, thus. 
7 in 789. 

As the stars in the heavens, the thousands of 
planets, the sun and the moon attest the operation 
of a Divine hand governing and controlling all things 
so these ten apparently insignificant figures have 
symbolism and uses that stamp them as coin coming 
from God’s laboratory. 

As this earthly planet is built upon and has at 
its center an ever-enlarging vacuum that eternally 
holds it in place and existence so the ten digits begin 
with an O, a vacuum, or, as it were, a nothing. 

As the vacuum is the most powerful of conceiv¬ 
able creations; as it is the only one thing that can 
exist on nothing and since it exists alone and with¬ 
out it none else can exist; and as it is more import¬ 
ant than the nucleus or the protoplasm and finally as 
it is the only conceivable creation capable of being 
at the center, where alone quietude absolutely exists, 
and- hence as it is the abiding place of Deity it is 
represented in numbers by the zero mark. 

As this little round figure that standing alone is 
without value it, and like Deity, seeks to give all 
itself to the environment so that all other than itself 
is one; all else in numbers come, from 0 and 1. 

This one is the one, the only one and sole mani¬ 
festation of its Creator. It is unchangeable, in¬ 
divisible, the same yesterday, to-day and forever. 
It can part itself into infinitude and multiply itself 
into eternity. 


56 


The 

number 

2 is 2 

ones 

ft 

tt 

3 ” 3 

tt 

a 

ft 

4 ” 4 

tt 

ft 

ft 

5 ” 5 

tt 

tt 

tt 

6 ” 6 

a 

tt 

tt 

7 ” 7 

a 

tt 

tt 

00 

00 

a 

tt 

ft 

9 ” 9 

a 


2 

3 

4 

1. 

— = 1. 

— = 1. 

-= 


2 

3 

4 

5 

6 

7 


— = 

= 1. — = 

= 1. — 

= 1. 

5 

6 

7 



8 9 

8 9 

One-half is a part of 1, but not of the One: it is 
J* of the 2’s 1. 

y 3 is one third of the 3’s 1. 1/6 is one sixth of 

the 6’s 1 

Yi is one fourth of the 4’s 1. 1/7 is one seventh 

of the 7’s 1. 

1/5 is one fifth of the 5’s 1. 1/8 is one eighth of 

the 8’s 1 

1/9 is one ninth of the 9’s 1. 

Composite units are combinations of the ten 
primes. 

The numbers 0, 2, 4, 6 and 8 are, as it were, and 
for special, as yet, unrevealed reasons, feminine 
numbers and 1, 3, 5, 7 and 9 are masculine numbers. 


57 


Each prime unit contains a symbolism that carries 
it into the world of the imponderable elements. 

0 is Divinity 

1 ” unity 

2 ” union 

3 ” trinity 

4 ’’ fortitude 

5 ” ability 

6 ” endurance 

7 ” retrospection 

8 ” providence 

9 ” perfection. 

The following is interesting and profitable 
13 = 3X3= 9 = 19 = 9 

2 3’s = 6X3 = 18 = 2 9’s = 18 

3 3’s= 9 X 3 = 27 = 3 9’s = 27 

4 3’s= 12 X 3 = 36 = 4 9’s = 36 

5 3’s = 15 X 3 = 45 = 5 9’s = 45 

6 3’s = 18 X 3 = 54=6 9 J s = 54 

7 3’s = 21 X 3 = 63 = 7 9’s = 63 

8 3’s = 24 X 3 = 72 = 8 9’s = 72 


> 3’s = 27 X 3 = 81 

= 9 9’s = 81 

9 = 9 

90 = 9 

18 = 9 

81 = 9 

11 

72 = 9 

36 = 9 

63 = 9 

45 = 9 

54 = 9 

72 = 9 

27 = 9 

81 = 9 

18 = 9 


09 = 9 


58 


The above put into Cho gives the following: 


bi 

fi 

ke 

fi 

su 

fi 

ke 


Pi 

ke 

bi 

pi 

ke 


pi 

di 

fi 

ke 

li 

su 

fi 

ke 

bi. 

ni 

ke 

di 

pi 

ke 

bi 

ni 

fi 

fi 

ke 

pi 

su 

fi 

ke 

di 

mi 

ke 

fi 

pi 

ke 

di 

mi 

gi 

fi 

ke 

ad 

su 

fi 

ke 

fi 

li 

ke 

gi 

pi 

ke 

fi 

li 

ki 

fi 

ke 

ak 

su 

fi 

ke 

gi 

ki 

ke 

ki 

pi 

ke 

gi 

ki 

li 

fi 

ke 

an 

su 

fi 

ke 

ki 

gi 

ke 

li 

pi 

ke 

ki 

gi 

mi 

fi 

ke 

eb 

su 

fi 

ke 

li 

fi 

ke 

mi 

Pi 

ke 

li 

fi 

ni 

fi 

ke 

eg 

su 

fi 

ke 

mi 

di 

ke 

ni 

pi 

ke 

mi 

di 

pi 

fi 

ke 

em 

su 

fi 

ke 

ni 

bi 

ke 

pi 

pi 

ke 

ni 

bi 


pi ke pi 


pio ke pi 


an ke pi 


nib ke pi 


em ke pi 


mid ke pi 


il ke pi 


af ke pi 


ok ke pi 


ug ke pi 


mid ke pi 


em ke pi 


nib ke pi 

pi ke pi 

an ke pi 


Also: 




0X5=0 

O 

su ki ke O 


1X5=5 

bi 

su ki ke ki 


2 X 5 = 10 

di 

su ki ke bi 

O 

3 X 5 = 15 

fi 

su ki ke bi 

ki 

4 X 5 = 20 

gi 

su ki ke di O 

5 X 5 = 25 

ki 

su ki ke di 

ki 

6 X 5 = 30 

li 

su ki ke fi 

o 

7 X 5 = 35 

mi su ki ke fi 

ki 

8X5 = 40 

ni 

su ki ke gi 

0 

9 X 5 = 45 

pi 

su ki ke gi 

ki 







59 


Also: 


0 x 7= 0 

0 

su 

mi 

ke 0 


3 X 7 = 21 

fi 

su 

mi 

ke di 

bi 

6 X 7 = 42 

li 

su 

mi 

ke gi 

di 

9 X 7 = 63 

pi 

su 

mi 

ke li 

fi 

2 X 7 = 14 

di 

su 

mi 

ke bi 

gi 

5 X 7 = 35 

ki 

su 

mi 

ke fi 

ki 

8X7 = 56 

ni 

su 

mi 

ke ki 

li 

1X7= 7 

bi 

su 

mi 

ke mi 

4 X 7 = 28 

gi 

su 

mi 

ke di 

ni 

7 X 7 = 49 

mi su 

mi 

ke gi 

Pi 

0X9=0 

0 

su 

pi 

ke O 


9 X 9 = 81 

Pi 

su 

Pi 

ke ni 

bi 

8 X 9 = 72 

ni 

su 

pi 

ke mi di 

7 X 9 = 63 

mi su 

pi 

ke li 

fi 

6 X 9 = 54 

li 

su 

Pi 

ke ki 

gi 

5 X 9 = 45 

ki 

su 

Pi 

ke gi 

ki 

4 X 9 = 36 

gi 

su 

pi 

ke fi 

li 

3X9 = 27 

fi 

su 

Pi 

ke di 

mi 

2 X 9 = 18 

di 

su 

Pi 

ke bi 

ni 

1X9 = 09 

bi 

su 

Pi 

ke pi 



All prime numbers have for their units digit either 
1 3 7 or 9 


11 

13 

17 

19 

31 

23 

37 

29 

41 

43 

47 

59 

61 

53 

67 

79 

71 

73 

97 

89 


83 








6o 


Any number not having one of the above four as 
its units digit is a composite number. Any composite 
number having one of the above four as its units 
digit must be divisible by either 3, 5 or 7. 

Prime numbers, with 1 as the units digit, from 
1 to 4001. 


1 

601 

1201 

2011 

2801 

3631 

11 

631 

1231 

2081 

2851 

3671 

31 

641 

1291 

2111 

2861 

3701 

41 

661 

1301 

2131 

2971 

3761 

61 

691 

1321 

2141 

3001 

3821 

71 

701 

1361 

2161 

3011 

3851 

101 

751 

1381 

2221 

3041 

3881 

131 

761 

1451 

2251 

3061 

3911 

151 

811 

1471 

2281 

3121 

3931 

181 

821 

1481 

2311 

3181 

4001 

191 

881 

1531 

2341 

3191 


211 

911 

1571 

2351 

3221 


251 

941 

1601 

2381 

3251 


271 

971 

1621 

2411 

3271 


281 

991 

1721 

2441 

3301 


311 

1021 

1741 

2521 

3331 


331 

1031 

1801 

2531 

3361 


421 

1051 

1831 

2551 

3371 


431 

1061 

1861 

2591 

3391 


461 

1091 

1871 

2621 

3461 


491 

1151 

1901 

2671 

3491 


521 

1171 

1931 

2711 

3511 


541 

1181 

1951 

2731 

3541 


571 



2741 

3571 





2791 

3581 



6i 


Prime numbers with 3 as units digit from 3 to 
4003. 


3 

433 

1013 

1523 

2113 

2903 

13 

443 

1033 

1543 

2143 

2953 

23 

463 

1063 

1553 

2153 

2963 

43 

503 

1093 

1583 

2203 

3023 

53 

523 

1103 

1613 

2213 

3083 

73 

563 

1123 

1663 

2243 

3163 

83 

593 

1153 

1693 

2273 

3203 

103 

613 

1163 

1723 

2293 

3253 

113 

643 

1193 

1753 

2333 

3313 

163 

653 

1213 

1783 

2383 

3323 

173 

673 

1223 

1823 

2393 

3343 

193 

683 

1283 

1873 

2423 

3373 

223 

733 

1303 

1913 

2473 

3413 

233 

743 

1373 

1933 

2503 

3463 

263 

823 

1423 

1973 

2543 

3533 

283 

853 

1433 

1993 

2593 

3583 

293 

883 

1453 

2053 

2633 

3593 

313 

953 

1483 

2063 

2663 

3623 

353 

983 

1493 

2083 

2683 

3643 

373 




2693 

3673 

383 




2713 

3793 





2753 

3803 



- 


2803 

3823 





2833 

3833 





2843 

3863 


3923 

3943 

4003 


62 


Prime numbers with 7 as units digit from 7 to 
4007. 


7 

457 

1087 

1747 

2417 

3217 

17 

467 

1097 

1777 

2437 

3257 

37 

487 

1117 

1787 

2447 

3307 

47 

557 

1187 

1847 

2467 

3347 

67 

577 

1217 

1867 

2477 

3407 

97 

587 

1237 

1877 

2557 

3457 

107 

607 

1277 

1907 

2617 

3467 

127 

617 

1297 

1987 

2647 

3517 

137 

647 

1307 

1997 

2657 

3527 

157 

677 

1327 

2017 

2677 

3547 

167 

727 

1367 

2027 

2687 

3557 

197 

787 

1427 

2087 

2707 

3607 

227 

797 

1447 

2137 

2767 

3617 

257 

827 

1487 

2207 

2797 

3627 

277 

857 

1567 

2237 

2837 

3677 

307 

877 

1597 

2267 

2857 

3697 

337 

887 

1607 

2287 • 

2897 

3727 

347 

907 

1627 

2297 

2917 

3767 

367 

937 

1637 

2347 

2927 

3797 

397 

947 

1657 

2357 

2957 

3847 


967 

1667 

2377 

3037 

3877 


977 

1697 


3067 

3907 


997 



3137 

3917 


3167 3937 

3187 3947 

3967 
4007 



63 


Prime numbers with 9 


4019. 

19 

509 

1319 

29 

569 

1399 

49 

599 

1409 

59 

619 

1429 

79 

659 

1439 

89 

709 

1459 

109 

719 

1489 

139 

739 

1499 

149 

769 

1549 

179 

829 

1559 

199 

839 

1579 

239 

919 

1609 

269 

929 

1619 

349 

1009 

1669 

359 

1019 

1699 

379 

1039 

1709 

389 

1049 

1759 

409 

1069 

1789 

419 

1109 

1879 

439 

1129 

1889 

449 

1229 


479 

1249 


499 

1259 



1279 

1289 



as units digit from 19 to 

1949 

2609 

3209 

1979 

2659 

3229 

1999 

2689 

3259 

2029 

2699 

3299 

2039 

2709 

3319 

2069 

2729 

3329 

2089 

2749 

3359 

2129 

2789 

3389 

2179 

2819 

3449 

2229 

2879 

3469 

2239 

2909 

3699 

2269 

2939 

3529 

2309 

2969 

3539 

2339 

2999 

3559 

2389 

3019 

3659 

2399 

3049 

3709 

2459 

3079 

3719 

2539 

3089 

3739 

2549 

3109 

3769 

2579 

3119 

3889 


3169 

3919 

3929 



3989 

4019 


6 4 


ORDERS. 

It is to be regretted that teachers of arithmetic 
talk of a square and of a cube root. 

In the whole realm of creation there is no such 
mathematical element as a square or a cube root. 
There is a root of units, a root of squares and a 
root of cubes, but there is not, and never will be, a 
mathematical square or a cube root. 

Wentworth and Hill say: “ The square root of a 
number is one of its two equal factors.” 

It is unscientific to make .one mathematical ele¬ 
ment applicable to two different symbols: the result 
is confusing. A factor is one of two or more num¬ 
bers that are employed in forming a product. To 
call it a square or a cube root is wholly unscientific. 
The same able authority says: “A cube root of a 
number is one of the three equal factors of the 
number.” 2 is one of the two equal factors of 4: 
it is also one of the three equal factors of 8 : hence, 
according to the definition, 2 is both a square and a 
cube root—which is ridiculous. 

To tell a pupil to find the square root of a certain 
number when no such thing exists is to add doubt 
and confusion to ignorance and leaves the novice 
duller than when he began his journey in the pur¬ 
suit of knowledge. One can easily find the root of 
a square, since the number itself should tell its root. 
Why complicate difficulties? 


65 


Limitations of arithmetic are fixed or should be 
fixed to the consideration of units, squares and 
cubes. To go beyond these is to get into deep 
water before knowing how to swim. To attempt 
to impart knowledge that is beyond the pupil’s com¬ 
prehension and ability of reception is to speak folly. 

These three classes, units, squares and cubes are 
as clearly defined and separated as are lines, squares 
and circles in geometry. All future success as a 
mathematician depends on the clear understanding 
regarding these demarcations. 

One serious and vital mistake (and fortunate had 
it been for humanity if it were the only one) made 
by instructors in the science of numbers is in im¬ 
plying that because a square is a product of two 
equal factors and the cube is the product of three 
equal factors that the several factors of units, 
squares or cubes are, in any respect, alike. Units 
can be squared and cubed, squares are restricted, as 
are cubes; each to its own order. 

Units are units; not squares nor cubes. 

Squares are squares, not units nor cubes. 

Cubes are cubes and are neither units nor squares. 


66 


UNITS. 

Units are entities. There are units of squares 
and units of cubes as there are innumerable entities. 

Units can exist of themselves but squares or cubes 
cannot have existence without units. 

Units can be added, subtracted, multiplied or di¬ 
vided in any order, or in any manner, while squares 
and cubes must follow a prescribed rule. 

Units are as the spirit of life. They come, we 
know not whence; they go, we know not whither, 
yet always existing, pliable and obedient to any 
will; to lie with the liar and to confirm the truth for 
the man of honor. To say that 2/5 X 7/9 = 
14/45 is to utter a lie. The figures are made in ig¬ 
norance and speak untruthfully. To say that 2/5 X 
7/9 = 1-8/45 is to utter a truth that will endure any 
trial. 

It is inexcusable to multiply a unit by a square 
or a cube, thus: 5 X 2 2 = 20, or 5 X 2 3 = 40. 

There are various forms of units among num¬ 
bers. There are prime numbers or those that can¬ 
not be exactly divided and which always (the num¬ 
ber 5 alone excepted) have for the units or right 


6 7 

hand digit, a 1, 3, 7 or 9 and there are units of 
composite numbers. 

A digit standing alone is a unit, thus 1, if it has 
another digit to its left, the one at the right is units, 
and the one on its left is the tens units thus: 12, 2 
units and 1 tens units; then follows the hundreds 
units 123, followed by the thousands 1234, and con¬ 
tinued indefinitely. 

Units have orders and degrees as surely as do 
squares and cubes. 

Taking multiples of units we have the following 
as the units digit. 

1 

2 4 

2 8 2, 4, 6, 8 = 20 * 

2 ( 1 ) 6 

2 (3) 2 

1 

3 3 

3 9 

3 (2) 7 1, 3, 7, 9 = 20 

3 (2) 1 

1 

4 4 

4 ( 1 ) 6 4 , 6 = 10 

4 (2> 4 

1 

5 5 

5 ( 2 ) 5 


= 5 


68 


1 

6 6 =6 

6 ( 3 ) 6 

1 

7 7 

7 (4) 9 1, 3, 7, 9 = 20 

7 (6) 3 

7 (4) 1 

1 

8 8 

8 (6) 4 2, 4, 6, 8 = 20 

8 (3) 2 

8 ( 1 ) 6 

1 

9 9 1, 9 = 10 

9 (8) 1 

Hence any number having the ’ following in the 
upper line as a units digit is divisible by the number 


beneath it. 

0* 1 2 

3 

4 

5 

6 

7 

8 

9 

5 3 4 

7 

2 

5 

2 

3 

2 

3 

7 6 

9 

6 


4 

9 

4 

7 

9 8 


8 


8 





provided the number is not a prime, in which case 
it is wholly indivisible and must have as the units 
digit a 1, 3, 7 or 9. 


6 9 


Even, or feminine, factors, 0, 2, 4, 6, 8 give an 
even or feminine units digit in the product, making 
it a feminine product. 

Even factors with odd or masculine factors. 0 
and 1, 2 and 3, 4 and 5, 6 and 7, 8 and 9, give a 
feminine product. 

Odd, or masculine, factors as 1, 3, 5, 7, 9 give an 
odd or masculine units digit in the product. 


70 


SQUARES. 

The confusing units, squares and cubes (wholly 
dissimilar quantities) in one measure, as is done by 
each and every arithmetic extant, is sufficient to 
bring condemnation upon all the present existing 
science of mathematics. 

Squares cannot be (by whatsoever necromancy 
figures are manipulated) made to become cubes : nor 
can cubes be changed into squares. Yet mathemati¬ 
cians persist in driving their thick skulls against this 
impasse. 

2 + 2 = 4. 

4 is the square of the root 2. 

2X2 = 4. 4 + 4 = 8 . This 8 is not a cube 

of the root 2. It is two squares. 

2 X 2 X 2 = 8. This 8 has nothing to do with a 
4. It is not and never was a square. This 8 is 
wholly different from the square’s 8; this one is a 
cube 8 from the root 2. 

4 + 4 = 8 = two squares. 

8 X 8 = 64 = two cubes. 

4^+ 4 + 4 = 12 = 3 squares. 

8X8X8 = 512 = 3 cubes. 

By the way what is a mathematical square ? 

It is the product of two equal factors. 


7i 


What is a root of a square.? 

It is one of the two equal divisors of any number. 

What is a mathematical cube? 

It is the product of three equal factors. 

What is a root of a cube? 

It is one of the three equal divisors of any num¬ 
ber. 

In fact there is so little in common between units, 
squares and cubes that no comparison can be es¬ 
tablished. 

Units are multiples of one. 

Squares are surfaces. 

Cubes are space. 

What is there in common with these three? 

A square 16 can be formed from the root 4; but 
16 from the root 2 makes 3 squares. 8 is two 
squares from the root 2, and 32 is 2 squares from 
the root 4. 

Hence it follows that a new mathematical nomen¬ 
clature should be formed to distinguish units from 
squares and cubes and each from the other. 

Regardless of the wonderful virtues, cleverness 
and simplicity of the Clio language herewith pre¬ 
sented the only feasible way for making these neces¬ 
sary distinctions is the use of words similar to Cho. 

Displace the 1 in each prime unit and insert an a 
to make that prime a root of a square or insert an a 
to make it a root of a cube. Add an a to a com¬ 
posite to make it a root of a square or an a to make 
it a root of a cube. 


72 


bi 

1 


la 

6 

root cu. 216 

di 

2 


mi 

7 


da 

2 

root sq. 4 

ma 

7 

root sq. 49 

da 

2 

” cu. 8 

ma 

7 

” cu. 343 

fi 

3 


ni 

8 


fa 

3 

root sq. 9 

na 

8 

root sq. 64 

fa 

3 

” cu. 27 

na 

8 

” cu. 512 

gi 

4 


nida 

8 

cu. of root 2 

ga 

4 

root sq. 16 

pi 

9 


ga 

4 

” cu. 64 

pa 

9 

root of sq. 81 

gida 

4 

sq. of root 2 

pa 

9 

” cu. 729 

ki 

5 


pif 

9 

sq. of 3 

ka 

5 

root sq. 25 

ao 

10 


ka 

5 

” cu. 125 

aoa 

10 

root sq. 100 

li 

6 


aoa 

10 

” cu. 1000 

la 

6 

root sq. 36 

aoad 

10 

sq. of 100 

aoad 

10 

cu. of 100 

ana 

18 

sq. 324 

ab 

11 


ana 

18 

cu. 5832 

abe 

11 

root sq. 121 

ap 

19 


aba 

11 

” cu. 1331 

apa 

19 

sq. 361 

ad 

12 


apa 

19 

cu.6759 

ada 

12 

sq. 144 

eo 

20 


ada 

12 

cu. 1728 

eoa 

20 

sq. 400 

af 

13 


eoa 

20 

cu. 8000 

afa 

13 

sq. 169 

eb 

21 


afa 

13 

cu. 2197 

eba 

21 

sq. 441 

ag 

14 


eba 

21 

cu. 9261 

aga 

14 

sq. 196 

ed 

22 


aga 

14 

cu. 2744^ 

eda 

22 

sq. 484 

ak 

15 

. 

eda 

22 

10648 

aka 

15 

sq. 225 

ef 

23 




















73 


aka 

15 cu. 3375 

efa 

23 

sq. 529 



al 

16 . 

efa 

23 

cu. 12167 


ala 

16 root of sq. 256eg 

24 




ala 

16 “ cu. 4096 ega 

24 

sq. 576 



alag 

16 sq. root 4 

ega 

24 

cu.12824 


am 

17 . 

ek 

25 




ama 

17 sq. 289 

eka 

25 

sq. 625 



ama 

17 cu. 3913 

eka 

25 

cu. 15625 


an 

18 . 

ekad 

625 

sq. of root 25 



ekad 

15625 

cu. of root 25 

Units Squares 

Units Cubes 



1 

4 

1 

8 




2 

9 5 

2 

27 

19 



3 

16 7 

2 3 

64 

37 

18 


4 

25 9 

2 4 

125 

61 

24 

6 

5 

36 11 

2 5 

216 

91 

30 

6 

6 

49 13 

2 6 

343 

127 

36 

6 

7 

64 15 

2 7 

512 

169 

42 

6 

8 

81 17 

2 8 

729 

217 

48 

6 

9 

100 19 

2 9 

1000 

271 

54 

6 

10 

121 21 

2 10 

1331 

331 

60 

6 

11 

144 23 

2 11 

1728 

397 

66 

6 

12 

169 25 

2 12 

2197 

469 

72 

6 

13 

196 27 

2 13 

2744 

547 

78 

6 

14 

225 29 

2 14 

3275 

531 

84 

6 

15 

256 31 

2 15 

4096 

721 

90 

6 

16 

289 33 

2 





17 

324 35 

2 





18 

361 37 

2 





19 

400 39 

2 





20 

441 41 

2 











74 


Thus it is evident that units are elements of one 
dimension. 

Squares, of two dimensions: cubes, of three di¬ 
mensions. 

Just as well use a pig’s tail for a yard stick as to 
try to measure volumes or capacities by cubes. 

Yet students are told that cubes are for volumes 
and capacities! 

By heavens! if the sea were not full of fish it 
would be well to deluge the land by filling it with 
scientists. 

Cubes are for space and for nothing than space. 

An empty square tin box is as much a cube as 
would be a square block of steel; the one takes 
exactly as much space as the other. 

The special application of cubes in the ordinary 
affairs of life will be more fully illustrated in Part 
2 of Cho Momo. 


75 


DIME, 

Fractions. 

Arithmeticians generally seemed obsessed by a 
notion that some meaningless adjective should qual¬ 
ify a fraction. 

Why should a fraction be called “ common,” 
“ proper,” “ improper” or “vulgar?” 

One does not say a “ common” 1, 2, 3, 4 or 5, nor 
a “ vulgar” 6, 7, 8 or 9. 

A fraction stands in the integrity of its useful¬ 
ness entirely separate from other mathematical ex¬ 
pressions. 

It seems more than passing strange that of the 
millions of mathematicians none has pointed out the 
fact that whole numbers apply exclusively to one 
and its multiples while fractions have to do solely 
with parts of a special one, applicable separately to 
each individual fraction. 

The indivisible one is never divided nor multi¬ 
plied. It or He, if you care so to honor it, stands 
alone or if in the number 2,. the one is twice; in 
number 3, three times, etc. In fractions, for in¬ 
stance 1/2 (one half) the 1/2 is a part of the 2’s 
one, thus: 2/2 = 1, and 1/2 of this is taken; not 
1/2 of the Supreme One. 

1/-3 of 3/3, 1. 1/4 of 4/4, 1. 1/5 of 5/5, 1. 
1/6 of 6/6, 1. 1/7 of 7/7, 1. 1/8 of 8/8, 1. 
1/9 of 9/9, 1. 1/12 of 12/12, 1. 1/13 of 13/13, 

1. 1/14 of 14/14, 1. etc. 


76 


Each fraction has two distinct parts, namely: a 
denominator and a numerator. 

The denominator applies to that especial one that 
has in its numerator and its denominator quantities 
equal to this fractional denominator, thus: 1/2 is 
1/2 of 1, but it is of the 2/2’s 1. 

The numerator is never applicable to the 1 indi¬ 
cated by the denominator, but applies soltiy to the 
number of times its own denominator is taken and 
it has no other possible function. 

When a numerator is greater than its denomina¬ 
tor the figures cease to be a fraction because the 
numerator in that case applies, in some respect to 
other than its own denominator, thus: 6/3 is not a 
fraction; it is the whole number 2 written in the 
form of a fraction: in the case of 8/3 the fractional 
form is 2-2/3, a whole number and a fraction. 

What then is the right definition for a fraction? 

A fraction is the expression of parts of 1. 

The Cho language uses the cardinal for the ex¬ 
pression of the numerator and the ordinal for that 
of the denominator, thus: 


bi die 

di fie 

fi gie 

gi kie 

ki lie 

1/2 

2/3 

3/4 

4/5 

5/6 

li mie 

mi nie 

ni pie 

pi aoe 

ao abe 

6/7 

7/8 

8/9 . 

9/10 

10/11 

ab ade 

ad afe 

af age 

ag ake 

ak ale 

11/12 

12/13 

13/14 

14/15- 

15/16 

al ame 

an ane 

an ape 

ap eoe 

eo ebe 

16/17 

17/18 

18/19 

19/20 

20/21 


77 


SUME. 

Multiplication op Fractions. 

Of all the hideous atrocities of arithmeticians the 
multiplication of fractions surely deserves the crown 
for idiocy, for consummate silliness, for the quintes¬ 
sence of assininity that brays at highest heaven. 

They (all of them, everywhere) assert that 1/3 
X 1/5 = 1/15 of the whole. 

That is to say: 1/3 of one dollar, or 33 1/3 cents, 
multiplied by 1/5 of one dollar, or 20 cents, equals 
1/15 of one dollar or 6 2/3 cents! 

They say that 1/2 X 1/2 = 1/4. 

They say that 4 X 1/4=1. That is to say that 
one having 4 dollars multiplies it by a quarter of a 
dollar and loses three dollars! 

One high authority says: “Multiplication of frac¬ 
tions is the operation of taking one number as many 
times as the units in another, when one or both are 
fractions.” 

Is not the above about as clear as dense mud? 

Another equally high (or low) says : “To find the 
value (sic) of a fraction; or, to multiply a fraction 
by a fraction, is as: 4/5 multiplied by 2/3, which 
means 2/3 of 4/5 = 8/15.” 

The above is simply criminal. If it had been 


78 


uttered by a nontutored (if such word is admissible) 
savage: (no, the pardon of the savage is asked!) 
Let us examine the problem. 


4/5 X 2/3 = 4X8 = 12 
2 X 5 = 10 
22 

5X3 = 15 


1-7/15 


Which bears the greater probability of correctness, 
1-7/15 or 8/15? 

Remember that the example given above for the 
correct method for multiplying a fraction is the first 
time in the history of man. Yet how simple it is. 
Only why was it not seen before? 

2/3 of-4/5 is wholly a different problem and has 
nothing to do with 4/5 X 2/3. 

The same authority quoted above gives one more 
example, which is copied letter by letter, as follows: 

“(12/25 X 2 13/16 X 10/21 = 12/25 X 45/16 

X 10/21 =fm X Im X ;W = 9/14)” 

5 £ 7 

2 

The above redundant figures are as published in 
the arithmetic under consideration. 

There is not the least necessity, justification or 
sense in using cancellation among these figures. It 
is the original fractions 12/25, 2 13/16, 10/25 that 
are to be multiplied and not some others. 

Take the problem under intelligent analysis and 
see the result. 


79 


12/25 X 45/16 X 10/21 = 

12 X 16 X 21= 4032 
45 X 25 X 21 = 23625 
10 X 25 X 16= 4000 

31657 = 

25 X 16 X 21 = 8400 7 ’ 

hence 12/25 X 45/16 X 10/21 = 3-6/8. 

They say that 1/3 X 1/5 = 1/15 
Let us see by the right method. 

1/3 X 1/5 = 1 X 5= 5 
1 X 3 =_3 

8 = 8 
5X3 = 15 15 

1/3 of one dollar is 33 1/3 cents 
1/5 ” ” ” V 20 

1/15 ” ” ” ” 6 2/3 ” 

8/15 ” ” ” ”• 53 1/3 ” 

Which is most probably correct? 

They say that 4X1/4 = 4/4 = 1; that is to say 
that if one has 4 dollars and multiplies it by 1 quarter 
of one dollar he loses 3 dollars. Nice arithmetic 
that would be surely. 

It has never been previously observed by any 
author that the multiplication by fractions have at 
least five different methods of procedure. 

Case 1. Is where a fraction is multiplied by a 
fraction, as: 2/5 X 7/9. 

No words of execration can express the utter con¬ 
tempt for those mathematicians (and the world 





8o 


seems full of such fools) who promulgate the fol¬ 
lowing : 

“Multiply the numerators together” (how can 
one multiply them separately?) “for a new numera¬ 
tor, and the denominators together (sic) for a new 
denominator” thus: they say, “2/5 X 7/9 = 14/45.” 
What have the whole numbers 2 and 7 to do with 
the fraction 2/5 ? or 5 and 9 to do with 7/9 ? 

The one right way (until a better is found) is 
to take the following rule. 

Rule I. For the multiplication of fractions. 
Take the sum of the products of each numerator 
with all the denominators, its own excepted, to get 
the multiplied numerator, and take the product of 
all the denominators for the required denominator, 
thus: 


r = 1-8/45 


2/5 X 7/9 = 2 X 9 = 18 
7 X 5 = 35 
53 

5 X 9 = 45 1 
20/21 X 3/7 X 4/9 X 2/3 = 

20 X 7 X 9 X 3 = 3780 
3X21X9X3 = 1701 
4X21X7X3 = 1764 
2 X 21 X 7 X 9 = 2646 
9891 


21 X 7 X 9 X 3 = 3969 


= 2-31/63 


Case 2. Is where a fraction is multiplied by a 
whole number, as 1/4 X 4. 



8i 


Under the admitted premises that as the numera¬ 
tor increases, the denominator remaining unchanged, 
the fraction is multiplied we have: 

1/4X4 = 4/4 = 1 1/2X2= 2/2 = 1 

2/4X2 = 4/4=1 2/3X3= 6/3 = 2 

3/4 X 2 = 6/4 = 1-1/2 3/9X5 = 15/9 = 1-2/3 

4/4 X 2 = 8/4 = 2 4/5 X 5 = 20/5 = 4 

Rule II. A fraction multiplied by a whole num¬ 
ber is its numerator multiplied by the number. 

Case 3. Is where a whole number is multiplied 
by a fraction, as: 4 X 1/4 = 4-1/4 
4/1 X 1/4 = 4/1 X 1/4 = 4 X 4 = 16 

1 X 1 = J . 

1X 4= 1 i = 4 - 1 * / 4 

4 1/4 X 1/4 = 17/4 X 1/4=17 X 4 = 68 

1 X 4 = _4 

—— 4-1/2 
4X4=16 7 

1/4 X 1/4 = 1 X 4= 4 
1X4=4 


8 


= 1/2 


4 X 4 = 16 
9 X 5/6 = 9/1 X 5/6 = 9 X 6 = 54 

5 XI— 5 
59 

1X6=6 


= 9-5/6 


Rule III. A whole number multiplied by a frac¬ 
tion equals the number plus the fraction. 

F 


82 


Case 4. Is where a whole number is multiplied 
by a whole number and a fraction, as: 4 X 2-1/4 
Here is a case requiring careiulness. It was 
stated in case 3 that a whole number multiplied by 
a fraction equals the number plus the fraction, but 
that will not prevail when the whole number has 
already been used by the number to which the frac¬ 
tion is adjoined, thus: while 4 -f- 1/4 = 4 1/4. 
When the 1/4 is adjoined to a whole number only 
the fractional part can be taken. 

One cannot work this case as follows: 


r The product would be too great, 
and 1/4 of 4 = 1. 8 + 1 = 9. 
Again: 9 X 3-6/7. 


9 

3-6/7 


4 

2-1/4 

8 

4-1/4 

12-1/4 

It is 4 X 2 = 8 


27 

Now, one has to take 1/7 of the 9 = 1-2/7, which 
multiplied by 6 = 7-5/7. 27 + 7-5/7 = 34-5/7. 

Rule Iy. A whole number multiplied by a whole 
number and a Traction is the product of the two 
whole numbers plus the fractional part of the mul- 
tiplican. 





83 


One other problem remains. 

Case 5. Is where a whole number and a fraction 
are multiplied by a whole number and a fraction, as: 


4-2/3 X 2-1/5. 

4-2/3 

2-1/5 2/3 X 2 = 4/3, 1-1/3 

8 

1 (1/3) 1/5 of 4 = 4/5, 

_2 

11 

because 


4 = 16/4 -=-1/5 = 16/20 = 4/5 


2/3 X 1/5 =! 


2 X 5 = 10 
1X3=3 


13 


3 X 5 = 15 

1/3 + 4/5 + 13/15 = 5/15 + 12/15 + 13/15 
= 30/15 = 2. 

Hence, 4 - 2/3 X 2 - 1 / 5 = 11 . 

Rule V. When a whole number and a fraction 
are multiplied by a whole number and a fraction, 
the product is the multiplican multiplied by the 
whole number in the multiplier plus the fraction in 
the multiplican multiplied by the whole number in 
the multiplier plus the fractional part of the fraction 
in the multiplier of the whole number in the multi¬ 
plican plus the product of the two fractions. 

Has any arithmetic prior to the present publication 
dared to enter into a thorough exposition of multi¬ 
plication of fractions? 



8 4 


They say: “14 X 3/7 = 6,” and give the fol¬ 
lowing remarkable illustration, 

“7)14 


and 


2X3 = 6” 


“14 

3 


7)42 

6 " 

Could any thing be more crude ? 

It has been shown above, in rule II. that “a whole 
number multiplied by a fraction equals the number 
plus the fraction,” hence the above example works 
itself. 

14 X 3/7 = 14-3/7 
Prove it: 14/1 X 3/7 = 14X7= 98 
3X1= 3 

= 14-3/7 
1X7= 7 d/ ' 

By what right did they afflict humanity with such 

a blunder as “14 X 3/7 = 6” ? 

They say “7/8 X 3/5 = 21/40” 

Do they? 

7/8 X 3/5 = 7 X 5 = 35 
3 X 8 = 24 

59 

8 X 5 = 40 = 1 ' 19 / 40 


85 


They give this display of stupidity: 

“7/11 x 11/21 = w % x = I /#* 

3 

What has cancellation to do with this job? It is 
none of its business. 

7/11 X 11/21 = 7 X 21 = 147 
11 X 11 = 121 

Hundreds of similar examples can be given were 
it not for idly filling space. 


86 


RECIPROCALS. 

Whence came this—this—(Oh, for words ca¬ 
pable of painting idiocy in true colors)—execrable 
rot? 

Reciprocals! 

Of what? 

Why does a fraction need a reciprocal? 

To enable fools to divide a fraction wrongly. 

They say that 9/7 is a reciprocal of 7/9! ! 

Why? 

They did not know how to divide 8/9 by 7/9, 
hence they turned the 7/9 upside down and made it 
stand on its head, as 9/7, and said “8/9 -r- 7/9 = 
8/9 X 9/7 = 72/68 = 1 9/63”! ! ! 

Writing of reciprocals one of the standard arith¬ 
metics of this age prints the following atrocity. 

“The reciprocal of a number is 1 divided by the 
number, thus: the reciprocal of 4 is 1/4, for (sic) 
4 X 1/4 = 1. The reciprocal of a fraction is the 
fraction with its terms interchanged, thus: 7/3 is 
the reciprocal of 3/7, for (sic) 7/3 X 3/7 = 1.” 

Can any conceivable thing be worse than the 
above quoted words? A fool purposely practicing 
folly for generations, finally believing folly to be 
truth, could not produce an offspring of greater 
idiocy. Yet millions of teachers and ten millions 
of scholars have stumbled over that barbarism. 

Once upon a time a master said to his slave, 


87 


Aesop: “ Go thou to the bath and see how many 
men are there. Bring me word that I can bathe and 
be clean.” 

Aesop sat at the entrance while a vast number 
of people pushed through an obstructed passage to 
the bath. Many were injured, some fatally, and 
few reached the cleansing waters. Aesop saw a 
great flinty rock in the middle of the path leaving 
only a small passage way on either side. Inquiring 
he learned that the rock had been placed there by 
powerful, ancient Magi and no man had been found 
capable of removing it. 

“ How long has it been here?” he asked. 

“ Since the time Cain slew his brother Abel,” was 
the answer. 

Aesop then by the might of his shoulder rolled 
the rock away and the crowd rushed in and out 
without thought and without a word of thanks. 

Coming to his master he said: “ Gracious lord 
and sovereign, I found but one man at the bath.” 

Then the master hastened. 

Reaching the bath he found a gay, happy multi¬ 
tude. 

“ What meanest thou, slave ?” 

“ Oh, master, when I came I sat at the door and 
saw the crowd fighting for entrance, none attempt¬ 
ing to remove the rock that stood in the way. Then 
came a man who looked, acted and took away the 
stone that had lain there for many ages: and he I 
thought to be the only one man.” 


88 


“ Who and where is he that did this great thing?” 
inquired the lord. 

“ Let your slave, most gracious master, whisper 
a word. I think I know the fellow. He, as I, is 
merely a slave. If my lord were to inquire for him 
he will presume and think himself greater than my 
master.” 

“Aesop,” the master said, “ you are right and I 
was wrong. Let the fellow go. Do thou, slave, 
accompany me, and cleanse me.” 

What is a reciprocal? 

It is a wild, undisciplined Hottentot captured by 
a ruse in a Brazilian jungle of South Africa and 
made to serve by forever standing on his head to 
divide a fraction, thinking the farther he separates 
his feet the better he has performed the task. 

If the word “reciprocal” were not tarnished by 
association with ignorance it possibly could be ac¬ 
cepted to express an important associate of each 
fraction to which associate Cho now gives the name 
Koad. 7/9 has the koad 2/9. 2/9 has the koad 7/9. 

7/9 + 2/9 = 9/9 = 1 
. 2/9 + 7/9 = 9/9 = 1 
3/5 + 2/5 = 5/5 = 1 
1/9 + 8/9 = 9/9 = 1 


Each is a koad of the other. 


89 


TUME. 

Division of Fractions. 

If incisive language was used in criticising the 
deficiencies of mathematicians with reference to the 
multiplication of fractions, not less can be said, and 
the English language is not vigorous enough to say 
more, regarding their presentation of division of 
fractions. 

Surely it promises little hope for the accuracy of 
any science of the present day when the foundation 
of all (mathematics) is so rotten. 

One of the standard arithmetics says: “ Division 
of fractions is the operation of finding how many 
times one number is contained in another, when 
one or more (sic) are (sic) fractions.” 

Now what sense is in such words? 

Who can understand such vague, confused 
mixture of terms? What has one number con¬ 
tained in another number to do with fractions? 

The same authority asks: “ What is the quotient 
of 7/8 divided by 14/5?” and gives the following 
bosh: “7/8 14/5 = 7/8 X 5/14 = 35/112 = 

5/16.” 

In the first place 14/5 = 2-4/5. 

How can 7/8 be divided, by any process, by 
2-4/5 ? 

They say that this 5/14 is the reciprocal of 14/5. 


90 


14/5 has no such possible element as a reciprocal: 
nor, in fact, has any fraction such unnecessary 
mathematical fiction. 

The above, like its colleagues, gives the following 
nonsensical law for dividing fractions. 

“ Invert the terms of the divisor, cancel, and 
proceed as in multiplication.” 

Can words express the shame of such display of 
ignorance ? 

Yet governments sanction by permission and 
financial support the teaching to children such folly! 

Another recognized authority taught in the public 
schools has the following: “ Multiplying by the re¬ 
ciprocal of a number is the same as dividing by that 
number; therefore, to divide by a whole number or 
a fraction, thus: 

“3/5 -r- 4 = 1/4 of 3/5 = 1/4 X 3/5 = 3/20.” 
and 

“3/5 -r- 7/8 = 8/7 of 3/5 = 8/7 X 3/5 = 
24/25.” 

If the above were not reprinted in the exact words 
used in the original publications our words could 
give little conception of their heinousness. 

The last named authority adds: Divide 14/17 by 
7, and gives 2/17 as its answer. 

It says: “It is evident that the fraction 14/17 is 
divided by dividing its numerator by 7, since the 
size of the parts, as denoted by the denominator, 
remains the same, while the number of parts taken 
is only 1/7 as large as before.” 


9i 


Who can understand such vague, unintelligible 
language ? 

In every arithmetic published the student is re¬ 
quired to divide a fraction by a whole number, as 
in the example. 

“3/5 -*-4 = 1/4 of 3/5 = 1/4 X 3/5 = 3/20.” 

Any mathematician of even mediocre ability 
should know that there are no arithmetical figures 
capable of dividing a fraction by a whole number: 
nor, in fact, can any number be divided, arithmeti¬ 
cally, by a number greater than itself. One cannot 
divide 5 by 8 ; then more surely one cannot divide 
3/5 by 4. To be sure 3/5 of any one thing can be 
divided into 4 parts, but the 3/5 are, in that case, as 
though they were 1 . 

There is one case and only one where there is an 
apparent exception to the universality of the above 
statement that a lesser number can never be divided, 
arithmetically, by a greater; and since this one 
exception has never before been presented by any 
known mathematician it is given here as “something 
new under the sun.” 

Number 1 , only, of all the numbers, can be 
divided by any greater number. 

And this number 1 is not the one indivisible one. 

The 2’s 1 can be divided by 2 

” 3’s 1 ” ” ” ” 3 

” 4’s 1 ” ” ” ” 4 

” 5’s 1 ” ” ” ” 5 

and so through every possible number. 


92 


Recalling the admitted premises, as given under 
multiplication of fractions, that “ when the denomi¬ 
nator is increased, its numerator remaining un¬ 
changed, the fraction is divided,” we have: 

2/2 (1) -f- 2 = 2/4 = 1/2. 

3/3 (1) -T- 3 = 3/9 = 1/3. 

4/4 (1) -r- 4 = 4/1.6 = 1/4. 

5/5 (1) 5 = 5/25 = 1/5. 

16/16 (1) -r- 16 '= 16/256 = 1/16. 

Can 1/6 be divided by 2? 

No. The 1/6 can be halved by 2, and this 1/2 of 
1/6 is 1/12 of the whole, but it is not 1/12 of 1/6. 

Cho presents the following different cases for the 
division of fractions. 

Case 1. Is where a fraction divides a fraction, 
as: 7/9 2/5. 

Rule I. A fraction divides a fraction when the 
sum of their numerators and the product of their 
denominators are taken, thus: 

7/9 s- 2/5 = l + g = 9/45 = 1/5 

Case 2. Is where a whole number is divided by 
a fraction, as: 6 3/5. 

Thus: 

Rule II. A whole number is divided by a frac¬ 
tion when it is divided by the denominator and the 
quotient is multiplied by the numerator, thus: 

6 3/5 = 6/5 = 1-1/5 X 3 = 3-3/5, 

but when the whole number is less than the denomi¬ 
nator of the fraction, rule I has to be employed thus: 


93 


6--7/9= J + l = 13/9 = 1-4/9. 

Case 3. Is where a whole number and a fraction 
are divided by a fraction, as : 6-2/7 H- 7/9. 

Rule III. Divide the whole number by the frac¬ 
tion according to rule I, or II, and add the division 
of the two fractions, thus: 

6-2/7 --7/9 = 6-— 7/9 = (! + ^=13/9 = 1-4/9. 

-t x y 

2/7 7/9 = * + 1= 9/63 = 1/7. 

4/9 + 1/7 = 28/63 + 9/63 = 37/63. 

1 + 37/63 = 1-37/63. 

Case 4. Is where a whole number and a fraction 
are divided by a whole number, as: 

6-2/3 Hr 2. 

Rule IV. Divide the whole number of the divi¬ 
dend by that of the divisor, and add the koad of 
the fraction. 

6 H- 2 = 3. Koad of 2/3 is 1/3. 

Hence, 6-2/3 H- 2 = 3-1/3. 

Case 5. Is where a whole number and a fraction 
are divided by a whole number and a fraction, as: 
6-2/3 H- 2-5/7. 

Rule V. Divide the whole number of the dividend 
by that of the divisor, and add the division of the 
two fractions, thus: 

6-2/3 H- 2-5/7 = 6 h- 2 = 3 + (2/3 H- 5/7) 

1/3 = 3-1/3. 


94 


Here is an easy example that any student of 
ordinary skill should be able to solve: Divide 111 
by 5-1/2 that the quotient multiplied by 5-1/2 will 
reproduce 111. 

All of the above examples are presented in a new 
method which will have to be legalized. 

Are any, or all of them, exactly correct? 

Possibly, no, and in fact, probably, no. 

They are correct until someone proves their de¬ 
fects and presents a better. It may be the good 
fortune of the author of Cho Momo to do better, 
but the crown of reward will come to the one who 
shows a better way for doing anything; and the one 
who accomplishes this is the victor. 

The following examples are taken from standard 
school arithmetics. One says: 

“ (Ex. 1. Divide 14/17 by 7. Ans. 2/17.)” 

“ First operation.” 

14/17 -T- 7 = 2/17. 

“ It is evident that the fraction 14/17 is divided 
by 7 by dividing its numerator by 7, since the size 
of the parts, as denoted by the denominator, remains 
the same, while the number of parts taken is only 
1/7 as large as before.” 

Remember the above was not uttered by an 
acknowledged ignoramus. It was written by a 
master of arts. 

Since no fraction can be divided by a whole 
number the above example is a farce. 


95 


Divide 12/13 by 6/13. Ans. 2. 

It says: “Since the fractional units (sic) of the 
two fractions are of the same kind, it is evident that 
12/13 contain 6/13 as many times as 6 is contained 
in 12; 12 -5- 6 = 2. Therefore, when the fractions 
have a common denominator, (sic) the division can 
be performed as in whole numbers by dividing the 
numerator of the dividend by the numerator of the 
divisor.” 

The above is simply despicable. 

Let us properly divide 12/13 by 6/13. 

U x» = 18/169 

hence 12/13 -r- 6/13 = 18/169 
and how by any force of imagination can fractions 
less than 1 make the whole number 2? 

They give the following poisonous mess of pot¬ 
tage. 

“ Divide 7/10 by 7/62. Ans. 6-1/5.” 

“7/10 -f- 7/62 = jr/10 X 62 /jT = 62/10 = 

6-1/5.” 

As a matter of fact, 

7/10 7/62 = jo x 62 = 14 / 620 = 7 / 810 - 

And the following horrible dish: 

“ Divide 9/28 by 3/7 and 3/4” 

3 

“ 9/28 -v- 3/7 = ? / X 111 = 3/4.” 

4 


96 


As a matter of fact, 

9/18 3/7 = jg + ® — 12/126 = 2/61. 

Then one more miserable contortion. 

Divide 27-3/5 by 6. Ans. 4-3/5. 

6)27-3/5 
4 + 3-3/5 

3-3/5 = 18/5 Which multiplied 
by 6 = 18/30 = 3/5 

4 + 3/5 = 4-3/5. 

That is simply dreadful, especially that the 3/5 
cannot be divided by 6 in any manner. 



97 


THE MU EE 
THE MU LI 

The Least Common Denominator. 

Of all the silly contrivances concocted for the 
confusion of civilization the “ least common denomi¬ 
nator” is not the least. 

There is properly no such thing as a least common 
denominator. 

It is a mathematical law that only like units can 
be added or subtracted. 

2/7, 3/8, 4/9, 7/12 and 8/15, in their present 
form cannot be added. 

It is necessary that some method shall be con¬ 
trived for giving each of the above fractions a 
similar denominator without changing the value of 
any of the fractions. 

Fortunately a method has been found and it is the 
one untarnished gem of the old regime. 

If the numbers 7, 8, 9, 12 and 15 are taken, re¬ 
gardless of the fact that they are similar to those 
in the several fractions, we can, by multiplying their 
several divisors and primes, find a one number, and 
only a one number from these especial numbers, that 
will give a similar denominator for the several given 
fractions, thus: 

G 


98 


3)7, 8, 9, 12, 15 

2)7, 8, 3, 4, 5 

2)7, 4, 3, 2, 5 

2, 3, 5 

3.X2X2X7X2X 3 X 5 = 2520. This 2520 
is the one only multiple these numbers can produce. 

It in itself is not a denominator though it can be 
used as such. It is called by Cho the mu le of those 
several numbers. 

Now if this mu le, or 2520 is divided by 7 we 
have 360. 

It is a known and generally admitted law since it 
can be easily proven, that if the numerator and the 
denominator of any fraction are multiplied by the 
same number the value of the fraction is unchanged. 

If both terms of the fraction 2/7 are multiplied 
by 360 we have a fraction of equal value in 
720/2520. This 360 is the mu li of the fraction 
2/7. It has nothing to do with a least common de¬ 
nominator. If the fraction 3/8 is similarly treated 
by its mu li 315 we have 945/2520, so-also with the 
fraction 4/9 by its mu li 280, 1120/2520: the frac¬ 
tion 7/12, by its mu li, 210, gives 1470/2520: in 
8/15, the mu li 168 makes it 1344/2520. 

Not one of the beautiful, clear and grand results 
are due to a least common denominator and the 
dragging in of such useless expression is confusing. 

The method by which the above result is gained, 
so necessary for adding or subtracting fractions, is 





99 


the one treasure found in the heap of waste material 
of the present day arithmetics. The author of Cho 
Momo thankfully does it all the honor possible. 


IOO 


PUME 

Addition or Fractions. 

Surely fractions are fascinating. The more they 
are added the sooner they are ended. With whole 
numbers addition can continue eternally: m frac¬ 
tions, when the numerator and the denominator be¬ 
come identical, or when the numerator becomes 
greater (top heavy as it were) than its denominator, 
the fraction ceases. 

In fractions there are two terms, a numerator and 
a denominator. In multiplication of fractions all 
the terms mutually act; in division the denominators 
are multiplied and the numerators are added; in 
addition and in subtraction the values of the de¬ 
nominators are unchanged, only the numerators are 
affected: in addition added, and in subtraction sub¬ 
tracted. 

No two or more fractions of unlike denominators 
can be added. Their denominators must be made 
similar without injuring the value of the fractions, 
thus: 1/2, 2/3, 4/5, and 5/6 cannot be added un¬ 
til the denominator of each is made alike. 

Take a 2, 3, 5, and 6, we find a mule, thus: 

2) 2, 3, 5, 6 

3) 3, 5, 3 


5 


2X3X5 = 30. 




IOI 


We give to 1/2, the muli 15 and get 15/30, to 
2/3, the muli, 10, and find 20/30, 4/5 has the muli 
6, and 24/30 result. 5/6 has muli 5, and then we 
multiply both terms and get 25/30, hence 15/30 -f- 
20/30 + 24/30 + 25/30 = 84/30 = 2-4/5. 

There is nothing confusing in the addition of 
fractions. With the magical assistance of the mule 
and .the muli they flow spontaneously to their sum. 


102 


RUME. 

Subtraction or Fractions. 

The assistance of the wonderful muli makes sub¬ 
traction of fractions nearly as easy as subtraction of 
whole numbers. 

Under the law that only similars can be conjoined 
or separated, in subtraction, as in addition it is 
necessary to give the two fractions employed like de¬ 
nominators without destroying the value of the 
fractions. 

In addition any number of fractions can be added, 
but in subtraction only two fractions, the minuend 
and the subtrahend, are practicable. 

When but two fractions are engaged in addition 
or in subtraction finding the mule is not required. 

Since the number that is the denominator of the 
other fraction is the muli, thus: in 7/9 — 2/5, the 
5 of the 2/5 is the muli of 7/9, thus 7/9 X 5 = 
35/45, and the 9 of the 7/9 is the muli of 2 / 5 , thus: 
2/5X9 = 18/45: and 35/45 — 18/45 = 27/45. 
This is invariable for the subtraction of fractions, 
consequently the problem is exceedingly easy. 

Only one feature need be emphasized. That is 
when a fraction is subtracted from a whole number, 
thus: 3 — 1/5. The 1/5 is not subtracted from a 
supposed cipher, but from one of the ones in the three 
and this one has a fractional form to accord with 


io 3 


the fraction that acts as the subtrahend, thus: in 3 — 
1/5, the 3 becomes 2 and the 1 is 5/5. 5/5 — 1/5 = 
4/5, hence 3 — 1/5 = 2-4/5. In 9 — 3/8, one of 
the nine’s ones, becomes 8/8, and from this the 3/8 
are taken, thus: 8/8 — 3/8 = 5/8, hence 9 — 3/8 
= 8-5/8, and 16 — 5/12 = 15-7/12. 


104 


RATEO. 

RATIO. 

(Pronounced by Clio rateo that a fluent and uni¬ 
versal pronunciation of mathematical expressions 
shall prevail.) 

It is surprising that such a general misconception 
of this very useful agent exists. It is made a frac¬ 
tion by every writer though it has neither the form, 
the uses nor functions of a fraction. The scientific 
attempts to wrongly describe ratio would be amus¬ 
ing if they were not fatally injurious to human 
progress. 

One of the standard authorities blindly says: 
“ The relative magnitude of two numbers is called 
their ratio, when expressed by the fraction which 
the first is of the second.” 

Where is the sense in such expression? 

One number is larger or smaller than another, 
but what kind of mathematical equation is relative 
magnitudes of two numbers? Who can understand 
the meaning of the words, “ which the first is of the 
second”? 

Continuing it says: “The terms of a fraction are 
called the terms of the ratio: the numerator is called 
the antecedent, the denominator is called the conse* 
quent.” 

Why was not this surprising fact taught under 


105 

the head of fractions? No one previously knew that 
a numerator is an antecedent. What is the authority 
for such statement? Yet it is blindly repeated by 
every arithmetic. 

It says: “2/3 is (sic) often written 2:3.” How 
can a third rightly become a three? Nor can gram¬ 
matical two-thirds take a singular predicate. 

It says: “ 5/7 = 5:7.” Why? 

Can a seventh be in any sense a seven? 

Enough of that authority! 

Another authority says: “ The comparison by 
ratio is made by considering how often one number 
contains, or is contained in another, thus: the ratio 
of 10 to 5 is expressed by 2, the quotient arising 
from the division of the first number by the second, 
or it may be expressed by 5/10 = 1/2, the quotient 
arising from the division of the second by the first, 
as the second or the first number shall be regarded 
as the unit or standard of comparison. “ In gen¬ 
eral, of the two methods, the first is regarded as 
the more simple and philosophical, and therefore 
has the preference in this work. Which of the two 
methods is to be preferred is not a question of so 
much importance as has been supposed, since the 
connection in which ratio is used is usually such as 
to readily determine its interpretation.” 

Heavens! think of the verbal construction of the 
above. The printer’s devil should have been con¬ 
signed to a dark cellar with a blue pencil and orders 
to “ cut it out.” 


io6 


To be sure we have known, through instruction 
of the operation of division that 10 divided by 5 
equals 2, but we never knew that 10 divided by 5 
also equals 1/2. 1/2 of what is this 1/2? How did 

they manage to make two whole numbers 10 and 5 
a fraction 5/10? Did ratio perform that wonderful 
deed? 

Let us drop these mathematical instructors as in¬ 
competent to perform a simple duty and refer to 
our two standard dictionaries, hoping for better re¬ 
sults. 

Webster’s dictionary says: “The relation which 
one quantity or magnitude bears to another of the 
same kind. It is expressed by the quotient of the di¬ 
vision of the first by the second, thus: the ratio 
3 to 6 is expressed by 3/6 or 1/2; of a to b by a/b; 
or (less commonly) by the second term to make the 
dividend, as: b:a = b/a.” 

The condemnation of the mathematical authority 
carries the same sentence against Webster. 

How can any necromancy convert a six into a 
sixth? What possible bearing has a one-half with 
a three or a six? 3 and 6 are whole numbers, 3/6 
is a fraction. How can two dissimilar elements 
enter into comparison? 

Let us try the Century dictionary, hoping for 
some better explanation of ratio than is given by 
any of the above quoted authorities. 

The Century says: “ The relation between two 
similar magnitudes in respect to quantity: the rela- 


tion between two similar quantities in respect to how 
many times one makes so many times the other: 
there is no intelligible difference between a ratio 
and a quotient of similar quantities; they are simply 
two modes of expression connected with different 
association. But it was contrary to the old usage 
to speak of a ratio as a quantity—a usage leading 
to intolerable complications. Instead of saying that 
the momentum of a moving particle is the product 
of its mass into its velocity—a mode of expression 
both convenient and philosophical—the older writers 
say that the momenta of two particles are in com¬ 
pound ratio of their masses and velocities.” 

Now, think of that! 

Instead of saying Mohammed came to the moun¬ 
tain the older writers asserted the mountain went 
to Mohammed. 

What has momentum and velocity to do with 
ratio ? Ratio as given by the Century is bad enough 
without “ compounding” it. We have heard of “ in¬ 
nocuous desertude” and here the Century perpe¬ 
trates “intolerable complications.” 

Then it remains that the field is open for any one 
who can do better than the best that has been done, 
and the field is always open everywhere and in any¬ 
thing for such daring aspirant. 

Cho’s definition for ratio is: the proportion two 
or more numbers bear to each other as compared 
with number 1. 


Thus: 




108 

10 : 

5: 

:: 1 : 1/2 

5: 

10 

: 1:2 

6 : 

3 

: 1 : 1/2 

3: 

6 

: 1:2 


If this is not to be the accepted definition and 
those given by the quoted authorities is to receive 
approval as well erase ratio from the science of 
mathematics. 

If the crown of approval is to be given Cho Momo 
for the promulgation of this law for ratio humanity 
can aspire to preparation for great achievements. 



OTEOM, PERCENTAGE 

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01 a single hair on the head of any one of the 100 oteo. 

If any single one of the negative 100 oteo thinks to become a portion of the af¬ 
firmative 100 oteo then at that very moment some one of the 100 oteo affirmative 


is beginning to change place, thus keeping the 100 oteo of either side exactly bal¬ 
anced. 

If 100 men possess 80% of the wealth of the earth all the balance of humanity 


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50 ” ” ” 50/100 ” 1/2 ” 

100 ” ” ”100/100 ” 1 thing. 



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affirmative, or gainers, and the negative, or losers and they were but 200 people on 
earth; and to each person were given one dollar for one year the result at the end 
of the year would be, and actually forever is, as follows: 








100% affirmative; 100% nfgative; 

100 PERSONS 100 PFRSONS 


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add G%, and have $1.06 lose 6%, and have $.94 

2%, or 2 of the 100 will, in one year, 2%, or 2 of the 100 will, in one year, 
add 7%, and have $1.07 lose 7%, and have $.93 




1%, or 1 of the 100 will, in one year, 1%, or 1 of the 100 will, in one year, 
add 8%, and have $1.08 lose 8%, and have $.92 

1%, or 1 of the 100 will, in one year, 1%, or 1 of the 100 will, in one year, 


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ITEU. 

Interest. 

Oteo Iteu. 

Per cent Interest. 

Interest, iteu, is the money that money makes. 

It is a financial percentage of gain. 

Oteom prevails in all things, hence in money mat¬ 
ters as in all else. Percentage and interest are not 
the same. 

Interest is financial percentage, but percentage is 
not interest. 

There are various ways of making money. One, 
and the chief, is a government’s credit that allows 
it to stamp and manufacture money based on vari¬ 
ous taxes and revenues. The main object of gov¬ 
ernmental power is the creation of credit that will 
enable it to manufacture money that can become 
generally accepted. It has the monopoly in this 
product. Another is through nature’s gifts, for 
which seldom thanks are returned but which are 
powerful factors in money making, namely: pro¬ 
viding habitation, food and clothing. Any power 
that can furnish these three to the people is supreme 
over all. 

Another, is from the sale of merchandise, and is 
called profits. 

Another is the money made by professional and 
personal services, in fees, salaries and wages. An- 


other, comes from trades and physical labor. An¬ 
other, and not the least productive, is the eleemosy¬ 
nary methods, now being very skillfully practiced, 
and called charities, emoluments and endowments. 

Another, the most powerful, the most dangerous 
and the most cruel is the unfair and iniquitous ad¬ 
vantage capital takes over labor in the high rates 
of interest on money loaned, and in dividends on 
stock purchased by money. Money should in all 
fairness never be permitted under any circumstances 
to make more than ten per cent per annum in any 
possible transaction. 

The more money makes money the less other 
pursuits will make it. And as long as the capitalist 
possesses this tremendously unfair advantage over 
labor war between the two will rightly continue. 

If Cho Momo can help “ one of the least of these” 
to higher, purer, truer and nobler things it has filled 
its mission. 

Interest should be paid for one day as though it 
were a month, and thirty days should be allowed 
for the financial month regardless whether the cal¬ 
endar month is long or short. 

Interest should be paid on any portion of one dol¬ 
lar, even though it is only one cent, as though it were 
an entire one dollar. The time and labor for sepa¬ 
rate calculation for parts of one dollar require this 
allowance. 

Cho Momo presents the following table for in¬ 
terest. 


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This measure is used for distances, generally geographic or those that pertain to 
the surface of the earth. Geometrical distances have another scale. 


The generally accepted table is as follows: 

12 inches make 1 foot 

3 feet ” 1 yard 




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% 


fi hei (3 miles) ke ” bi kei (1 league) 

c/p de bi gie kei (69/4 leagues)ke ” bi lei (1 degree) 

fi ao lei (360 degrees) ke ” bi mei (1 great circle) 



The above can be arranged conveniently as follows: 


127 


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cd 

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finioaloo ke ibanoo ke aouloo ke bipidon ke ginio ke ao ke bi. mei 

3801600 =0 316800= 105600 = 19200 = 480 = 60 = 1. the great circle 

of the earth 










































1368576000 == 114048000 


i 


128 


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I 







129 


SQUARE MEASURE. 

The unpremeditated invective against the gross 
ignorance shown by every arithmetic published 
leaves language barren before the following aston¬ 
ishing table. 

The accepted table for square measure. 


144 

square 

inches make 

1 

square 

foot 

9 


feet 

>> 

1 


yard 

30 >4 

>> 

yards 


1 

>> 

rod 

40 

)> 

rods 

}} 

1 

rood 


4 


roods 

>> 

1 

acre 


40 

perches 


>> 

1 

acre 


640 

acres 



1 

square 

mile 


144 square inches make 12 square feet, not 1 
9 square feet make 3 square yards, not 1 
144 line, or ordinary inches make 1 square foot 
12 square inches make 1 square foot. 

9 yards make 1 square yard, but 9 square yards 
are 81 yards and they make 9 square yards, not 1. 

Why not say, equally wrongly, that 640 squarq 
acres make 1 square mile? 

i 


130 


By what right and authority are such blunders 
made and where is the discernment of governments 
that justify such mistakes? 

The above named table of “ square measure” is 
wholly wrong and is utterly useless. It should have 
no place in any scholar’s mind and should be ignored 
by all. The student that aspires to ability in mathe¬ 
matics has no space for such dull tools. 

Yet, not content with that egregious error, they 
add the following nonsensical meter table: 

A square millimeter = 0.000001 of a square meter. 
” ” centimeter = 0.0001 ” ” ” 

” ” decimeter =0.01 ” ” ” 

” ” meter —- Principal Unit. 

” ” dekcameter = 100 square meters. 

” ” hektometer = 10000 ” 

” ” myriameter = 1000000 

Observe the 0 placed at the left of a decimal 
point. The cipher so placed is expressive of nonen¬ 
tity, of imbecility, of inexcusable ignorance. A 
mathematician has no right to so use the 0 as above 
than a man of culture has the right to profane lan¬ 
guage by the expression “ who to,” instead of “ to 
whom.” 

The expression “ principal unit” is a technical, 
not a mathematical term, and is wholly incompre¬ 
hensible. They could not easily square 39.37, which, 
squared would give the horrible combination of 


I 3 I 

39.37 

39.37 

27559 

10911 

35433 

10911 


1459.0969 

There is not a possible mathematical problem 
that requires such extremely high or low numbers: 
and if required would not use a square metric sys¬ 
tem in the acquisition. 

Properly there is no such thing in mathematics 
as a square meter, any more than there is a square 
root. Any series of numbers can be squared with¬ 
out the worthless and cumbersome metric square. 

Let this go the way of other useless matter into 
oblivion. 




132 


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of purchase of surface. 

The following square, or surface measure is a necessity for conducting the ordi¬ 
nary affairs of life. 


12 square inches (144 inches) = 1 sq. foot 


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The above square, or surface measure is expressed in Cho language as follows: 








134 


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work a peer of the great Leuukenhoek who could measure the length of a joint of 
a minute insect’s leg, and who though living over one hundred years past has had no 
successor his equal, the scale for minute parts of an inch will be important. 


The Inch, or Cei Scale. 

12 marks = 1 point 

ad bui ke bi boi 


*35 



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THE CUBIC MEASURE. 


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G C 

rt CD 


3 c 


CD 


E £ 


CD 

C/3 

• ^H 

r"| 

£ 


O CO 

TH GG 

CD 

G 


CD 


2 > 
^ CO 
CO C3 


CO 

CO 

• r-H 

CD 

bjo 

gg 

CD 

CD 

C/3 

O 


CO 

o3 
CD CD 


GG 

G 

CD 

o3 


C 

o3 

co 

03 


Li 

CD 

4-> 

CD 


CD 

GG 


GG 

CD 

CD 


CO 

G 

O 

CD 

CD 

P 

G 

CD 

CD 

GG 


Li 

o3 

o. 


o3 to co 
CD O 




oJ 

co 


03 


CD 

4-> 

CD 

£ 


CD ~ 

GG 

^ •• CD 

^ OG 
.£2 vi3 +J 

co 


CJ 


£ £ 


03^0 

GG _2 




CD 


CD 

G 

O 

co 

• f-H 

GG 

G 

03 

f-H ^ 

^ *N 

CD 

h-> CD 
GG- bJO 
GG 
03 CD 


It is enough to make one dizzy! 

% 

Again this Cerberus, No. 3937, an emissary of Satan, the guardian of Hades, 
shows its fangs. One stronger than it is coming to capture, strangle and kill it. 









Yet not content with that emanation of diseased brains they give the following: 

1 milliliter == .061023 cubic inches 


i45 


o 3 

U 

O 


co 

cd 


u 

<v 


G 

cd 


u 

CD 


u 

<v 

4 -» 

<v 

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<u 

• r-H 

• r-H 

U 

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"O 

4 -» 

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• r-H 

/*—s 

C J 

<D 

U 

O 

O 

<v 

4 -> 

<D 

(V 

• r-H 

G 3 

• rH 

G 3 



rH 

rH 

rH 

rH 

rH 


in 

g; £ 

u 

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G 

rt <u 


c/) 

G 

• r-H 

G 

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cu 


s 

a3 

m 

<U 







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4—> 

03 

r 1 < 


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CD 

CD 

<U 

O 

cu 

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CD 

G 

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CU 

M-H 

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CD 

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rs 

rs 


»-n 

»\ 

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IS 

r\ 


CD 

03 

4 -* 

G 

<u 




• 


G 



M—l 

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CD 



O 

cu 

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CO 

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03 

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10 

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cn 

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£> 

OP 

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W 

Pi 

P 

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Q 

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CD 

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J 


“ Since” (they say) “ 1 bushel = 2150.4 cu. in. = 1 pk. = 537.6 cu. in. 
1 qt. = 67.2 cu. in. = 1 pt. = 33.6 cu. in. 
therefore: 


























1 milliliter == .001816 pints 

1 centiliter = .018161 ” 

1 deciliter ==s .181611 


146 


<v 

s- 

a 



05 

y—i 

rH 

CO 

T - 1 

OO 


S-H 

03 


Oh Oh 
05 05 


. c n 

. c n -*-> 

t±, h-* cr 

H C7 1 

o Oi ^ 
iO . • 

O in 
00 04 ^ 

9 ^ rH 

rH CO 


04 S 
Oh 02 


GO 

05 05 


i - 1 53 

• t-H < -j > 

02 O — 

CO H—i 

o o 

<V 03 
02 


04 


w 

C4 

D 

c/3 

< 

w 


0 

a 


a3 

o 

>% 

<v 


d 

r~| 

> 


m 

<v 

3 

a3 


w 

O 

.2 

’> 

a; 

J-h 

r>. 


k4 ^ — 

rH rH r—t ^ r 


in 


Pi 

O 

w 

£ 


w 

w 

Jh 


C/3 

Co 

in 

£ 

S-H 

a> 

- 4 -> 


rt 

C/3 


bJO 

CO ^ 
o CO 
h|H 10 
CO ^ 

o GO 

O' o 
• • 

o 


U 

03 


03 02 




o 


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03 


CL) 
C J 


C/3 


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O ^ 
P0 CO 
CO 


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CO 

10 

OD 


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Oh 

O' 

00 

CO 


C/3 


05 


C/3 


C/3 

__ 25 

a3 bio 

o 

05 05 


11 !i H li 


s_ 


Vh 

<u 

<L> 

4-> 


0/ 

H—> 

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- 4 —> 

• H 

• rH 

S-H 

25 

O 

-4—> 

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03 

0 

CJ 

JL3 

H-> 

03 

a> 

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• r—< 

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r-* 

rH 

rH 

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tH 


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o 


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r -1 

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r- 1 

55 




1 kiloliter, or stere = 1 tun, 12 gals. 0 qt. 1 pt. 1.44 gills.” 

Such horrible tables will do for the place whence Cerberus came, but they 
not suitable for this fair earth of ours. 














i47 


LIQUID OR WINE MEASURE. 


4 gills make 

1 pint 

2 pints 

>> 

1 quart 

4 quarts 

>> 

1 gallon 

63 gallons 

}> 

1 hogshead 

2 hogsheads 


1 pipe, or butt 

2 pipes 

>> 

1 tun 

Which put into Cho is 



gi ta 

ke 

bi te 

4 gills 

= 

1 pint 

di te 

ke 

bi ti 

2 pints 

= 

1 quart 

gi ti 

ke 

bi to 

4 quarts 

= 

1 gallon 

ai to 

ke 

bi tu 

63 gallons 

== 

1 hogshead 

di tu 

ke 

bi ta 

2 hogsheads 

= 

1 pipe 

di t a 

ke 

bi tan 

2 pipes 

= 

1 tun 


THE DRY MEASURE. 

2 pints make 1 quart 
8 quarts ” 1 peck 

4 pecks ” 1 bushel 


In Clio it is: 


148 

di te ke bi ti 

2 pints = 1 quart 

ni ti ke bi tim 

8 quarts = 1 peck 

gi tim ke bi tom 

4 pecks = 1 bushel 

MEASURE OE TIME. 


60 seconds 

make 

1 

minute 

60 minutes 

}) 

1 

hour 

24 hours 

n 

1 

day 

7 days 

a 

1 

week 

4 weeks 


1 

month 

12 months 

if 

1 

year 

365-1/4 days 

if 

1 

year 

in Cho language is: 
ao ha 

ke 

bi 

he 

60 seconds 

= 

1 

minute 

ao he 

ke 

bi 

hi 

60 minutes 

= 

1 

hour 

eg hi 

ke 

bi 

ho 

24 hours 

= 

1 

day 

mi ho 

ke 

bi 

hu 

7 days 

== 

1 

week 

gi hu 

ke 

bi 

ha 

4 weeks 

= 

1 

month 

ad ha 

ke 

bi 

ham 

12 months 

= 

1 

year 

fi ak de bi gio ho 

ke 

bi 

ham 

365 and 1/4 days 


1 

year 


149 


CIRCULAR MEASURE. 


60 seconds make 
60 minutes 
30 degrees 
12 signs 
This in Cho is: 

ao ha ke bi 
60 seconds =. 1 

ao he ke bi 
60 minutes = 1 

io leim ke bi 
30 degrees = 1 

ad sein ke bi 
12 signs = th 


1 minute 
1 degree 
1 sign 

the circle of the zodiac 
he 

minute 

leim 

degrees 

seim 

sign 

meim 

* circle of the zodiac 


THE AVOIRDUPOIS TABLE. 


This is a most important and useful scale. 


16 drams 

make 

1 ounce 

16 ounces 

33 

1 pound 

25 pounds 

33 

1 quarter 

4 quarters 

33 

1 hundred weight 

20 hundred weights 


1 ton 

The avoirdupois in the Cho language is as follows 

al la 

ke 

bi le 

16 drams 

= 

1 ounce 

al le 

ke 

bi li 

16 ounces 

= 

1 pound 

ek li 

ke 

bi lo 

25 pounds 

= 

1 quarter 

gi lo 

ke 

bi lu 

4 quarters 

= 

1 hundred weight 

eo lu 

ke 

bi 1 a 

20 hundred weights 

= 

1 ton 


Cho Momo herewith creates a new table, to be 
known as the Gom table, or scale. It refers to 
parts of a grain. 


ao gigom 
10 ” 
ao figom 
10 ” 
ao digom 
10 ” 
ao gom 
10 ” 

The Troy, or Mint 
radium, platinum, etc. 

It is as follows: 

10 gom 
24 grains 
20 penny weights 
10 ounces 

The above placed in 
ao gom 
10 ” 
eg goma 
24 grains 
eo lam 

20 penny weights 
ao lem 
10 ounces 


ke bi figom 
= 1 ” 

ke bi digom 
= 1 ” 

ke bi gom 

= 1 ” 

ke bi goma 
== 1 grain 

weight is for gold, silver, 

make 1 grain 

” 1 penny weight 

” 1 ounce 

” 1 pound 

Cho language is as follows: 
ke bi goma 
= 1 grain 

ke bi lam 
— 1 penny weight 

ke bi lem 
= 1 ounce 

ke bi lim 
= 1 pound 


THE GRAM TABLE. 

Arithmetical authorities give the following un¬ 
workable table, saying the base is one gram, and 
making the gram equal “ the weight of a cubic 


152 


centimeter of rain water,” which, they say, equals 
“ 15.432 grains, Troy, nearly,” and equals “.0352745 
ounces, avoirdupois nearly.” 

Why not have said: the cubic centimeter of 
moon-shine, or the one tenth millionth of the weight 
of the atmosphere lying between the equator and 
the north pole? 

What is the mathematical value of the word, 
“ nearly?” 

The Gram table is the following ridiculous mess. 


‘ equivalents in avoirdupois and troy weight. 


‘ 1 milligram 
1 centigram 
1 decigram 
1 gram 
1 decagram 
1 hectogram 
1 kilogram 
1 myriagram 
1 quintal 


0.0154 grains Troy 
0.1543 ” 

1.5432 ” 

15.4327 ” 

0.3527 ounce Avoirdupois 
3.5274 ” 

2.2046 pounds ” 

22.046 

220.46 


1 millier, or ton = 2204.6 


The above table is hereafter at auction to the 
lowest bidder. Cho Momo has no use for such 
stuff. 

A Coal, Sand, Brick, etc.. Scale. 


Any of the above should be calculated at a stan¬ 
dard of ten dollars ($10.00) for one ton of 2,240 
pounds. 


153 


If coal were at $5.00 per ton, the cost would be 
5/10 of any of the prices established in the follow¬ 
ing scale. 

The Cho word for ton, or tons, is la;; and that for 
pound, or pounds, is li. 


eddo 

li 

ke 

bi la 

a ao 

a 

2240 pounds 

= 

1 ton 

at $10. 

dollars 

ebeo 

li 

ke 

bi die la 

a ki 

a 

1120 

pds. 

= 

1/2 ton 

at $5. 

dollars 

kiao 

li 

ke 

bi gie la 

a di de bi die a 

560 

pds. 

= 

1/4 ton 

at $2.50 

dollars 

dinio 

li 

ke 

bi gie la 

a bi de bi 

gie a 

280 

pds. 

= 

1/4 ton 

at $1.25 

dollars 

bido 

li 

ke 

bi ale la 

a an 

a 

140 

pds. 

= 

1/16 ton 

at $.68 

cents 

mio 

li 

ke 

bi ide la 

a ip 

0 

70 

pds. 

= 

1/32 ton 

at $.39 

cents 

. ik 

li 

ke 

bi age la 

a eo 

0 

35 

pds. 

= 

1/64 ton 

at $.20 

cents 

an 

li 

ke 

bi di ene la 

a ao 

0 

18 

pds. 

= 

1/128 ton 

at $.10 

cents 

pi 

li 

ke 

bi di ube la 

a ki 

. 0 

9 

pds. 

= 

1/256 ton 

at $.5 

cents 

ki 

li 

ke 

bi ki ade la 

a fi 

0 

5 

pds. 

= 

1/512 ton 

at $.3 

cents 


54 


THE MONETARY OR OU SCALE. 

Part I of Cho Momo closes with the simplest, 
the grandest, the most serviceable and best scale of 
all, namely: the Ou Table. 

In comparison no other monetary table can equal 
that universally used by the people of the United 
States of America. Perhaps no other one element 
has been as instrumental in creating the wonderful 
prosperity of the people of that government as its 
simple decimal financial table: and positively, pro¬ 
viding the remarkable utility of this table is ad¬ 
mitted, no government can hope to survive in the 
struggle for existence that does not arrange its 
finance in accord with this simple scale. 

the monetary table. 

10 mills make 1 cent 
10 cents ” 1 dime 

10 dimes ” 1 dollar 

In the universal Cho "commercial language the 
above is: 

ao u ke bi 0 

10 mills = 1 cent 

ao o ke bi i 

10 cents = 1 dime 

ao i ke bi a 

10 dimes = 1 dollar 

di (apples) ta ki 0 
ki ta di 0 


155 


EPILOGUE. 

Cho Momo entered the arena finding the field 
crowded with wildly running contestants. 

One look swept the sands of all that rubbish. 

Cho Momo now leisurely walks alone challenging 
an equal, and ready to submit to a superior. 

He who can do better than Cho Momo herein 
presents has riches, honors and fame for rewards. 
Ability asks no permission: it dares and does. 

Study Cho Momo and get ability to dare and do. 


PRONUNCIATION. 


The author of Cho Momo literally copies the 
words given in Prof. Alex. Melville Bell’s Visible 
Speech as the highest and best authority on linguistic 
sounds extant. 


A 

as 

a 

in age 

as 

aye 

in 

aye 


}> 

ai 

” aim 

>> 

ea 

>> 

steak 


)} 

ao 

” gaol 

jj 

ei 


vein 



ua 

” guage 


ey 

>> 

obey 


>> 

ay 

” day 


eye 

jj 

preyed 

E 

as 

e 

in eve 

as 

eo 

in 

people 


>> 

ea 

” eat 

j." 

ey 


key 


)> 

ee 

” see 

5) 

eye 


keyed 


D 

e’e 

” e’en 

J) 

i 

>> 

fatigue 



ei 

” conceive 

>» 

ie 

)) 

field 

I 

as 

eigh 

in height 

as 

igh 

in 

nigh 



eye 

eye 


ui 


guide 


>> 

i 

” time 

)> 

uy 

jj 

buy 



ie 

” tie 

as ye 

in rye 

y 

>) 

by 

0 

as 

aut 

in hautboy 

as 

oa 

in 

oak 



eau 

” beau - 

j> 

oe 

5 J 

foe 


>> 

eo 

” yeoman 

j» 

00 


brooch 


■ ff 

eu 

” shew 


ou 


soul 



ewe 

” sewed 

>» 

ow 

)) 

crow 


>> 

0 

” old 


owe 

; « 

owed 


157 


u 

as eu 

in rheum 

as oo 

in bloom 


>> 

ew 

grew 

” ooe 

” wooed 


” ewe 

brewed 

” ou 

” through 


” o 

” do 

” u 

” rule 


” oe 

” shoe 

“ ue 

rue 


” oeu. 

manoeuvre 

” ui 

” fruit 



as wo in two 


o, 

o, as a 

in all 

as 

aw in saw 


” au 

” fraught, laud, naught ” 

awe ” awed 

A, 

a, as ar 

in hard 

as 

ear in heart 


” er 

clerk 

a 

uar “ guard 



as a in 

father 



In printing, an italic letter is used to express an 
underscored written character. 


C as c in cell 

” ce ” ice 

ps ” psalm 
” s ” sale 

G as gu in guide 


as se in science 
sch ” schism 
” se ” base 
” ss ” loss 


The other consonants are pronounced as com¬ 
monly used. 

A consonant makes one syllable with its preceding 
vowel, thus: ab; but if followed by a vowel it 
drops the first vowel and forms another syllable 
with the succeeding vowel, thus: a be. 

The accent is always on the first syllable with 
words of two or three syllables: and on the ante¬ 
penult with words of more than three syllables. 



159 


CHO VOCABULARY. 


A 

Dollars 

(D)a 

Root of square 

Ad 

Forms the possessive case 

Ae 

Billimeter 

Adil 

Abstract numbers 

Amomo 

Abstract arithmetic 

Aon 

All the fingers 

A 

At 

(P )“ 

Root of cube 

Adil 

Concrete numbers 

Ala 

Arabic 

A momo 

Concrete arithmetic 

Be 

Millimeter 

Bei 

Bei 

Bib 

Prime numbers 

Bid 

Composite numbers 

Bidil 

Masculine numbers 

Bii 

Bii 

Boi 

Boi 

Bu 

Parentheses 

Bua 

ist parentheses 

Bue 

2 d parentheses 

Bui 

3 d parentheses 

Buo 

4 th parentheses 

B^n 

Left index finger 


6 o 


Ce 

Cei 

Cho 

Cho E 
Chom 

Cho Momo 

Centimeter 

Inch 

One 

Meter 

Magnetism 

The one arithmetic 

De 

De 

Dem 

Dil 

Dile 

Dili 

Digom 

Dim 

Dime 

Dimi 

Dimo 

Dimu 

Don 

And 

Decimeter 

Linear measure 
Numbers 

Ordinal numbers 
Cardinal numbers 
Digom 

Whole numbers 
Fractions 

Numerator 

Denominator 

Mixed numbers 

Left long finger 

E 

Ei 

Ea 

Eom 

Eon 

Meter 

Yard 

Square yard 
Immaterial 

Material 

Fe 

Fei 

Fea 

Figom 

F o?i 

Decameter 

Rod 

Square rod 

Figom 

Left ring finger 



161 

Ge 

Gei 

Gea 

Gigom 

Gom 

Goma 

Gtfm 

Hectometer 

Furlong 

Square furlong 
Gigom 

Gom 

Grain 

Left little finger 

Ha 

He 

He 

Hei 

Hea 

Hi 

Him 

Ho 

Hu 

H a 

Ham 

Seconds 

Kilometer 

Minute 

Mile 

Square mile 

Hour 

Time 

Day 

Week 

Month 

Year 

I 

Ie 

Ike 

Iteu 

Dimes 

Myriameter 

Equivalent 

Interest 

Kal 

Ke 

Kei 

Ko 

Koad 

Kon 

What 

Equal, makes, is, are 
League 

Vacuum 

Co aid 

Right little finger 


K 


162 


K a 

Environment 

Kaom 

Electricity 

La 

Dram 

Lak 

Price, cost of anything 

Lam 

Pennyweight 

Le 

Ounce 

Lei in 

Degree 

Lem 

Ounce 

Li 

Pound 

Li in 

Pound 

Lo 

Quarter 

Lu 

Hundredweight 

Lu 

This, these 

Luad 

Their 

L<?n 

Right companion finger 

L a 

Ton 

Mei 

Circle of earth 

Meim 

Circle of zodiac 

Momo 

Arithmetic 

M<?n 

Right long finger 

Mu 

That, those 

Muad 

Their 

Mule 

Mule 

Muli 

Muli 

Muod 

Them, of them 

Nid# 

Cube 8 of root 2 

N^n 

Right index finger 

Not 

Naught 


Nu 

Units, unit 

Nua 

Square 

Nutf 

Cube 

0 

Center 

0 

Cents 

0 

Cipher, zero, naught 

Od 

Forms objective case 

Odil 

Feminine numbers 

On 

Imponderable* 

Oteo 

Per cent. 

Oteom 

Percentage 

Ou 

Money 

On 

Left thumb 

Pib 

Name of the Pib division 

Pibo 

ist section of the division 

Pido 

2 d section of the division 

Pifo 

3 d section of the division 

Pigo 

4 th section of the division 

Piko 

5 th section of the division 

Pilo 

6 th section of the division 

Pimo 

7 th section of the division 

Pino 

8 th section of the division 

Pipo 

9 th section of the division 

Pito 

10 th section of the division 

Fon 

Right thumb 

Pu 

Plus 

Pua 

Magnets 

Puine 

Addition of fractions 

Puum 

Addition 

Puo 

Sum 


6 4 


Rateo 

Ratio 

Ru 

Subtrahend 

Ru 

Minus 

Rui 

Minuend 

Ruo 

Difference 

Rume 

Subtraction of fractions 

Ruum 

Subtraction 

Seim 

Sign 

Su 

Multiplier 

Sua 

Factors 

Sui 

Multiplican 

Sume 

Multiplication of fractions 

Suo 

Product 

Suum 

Multiplication 

Ta 

Gill 

Te 

Pint 

Ti 

Quart 

Tim 

Peck 

To 

Gallon 

Tom 

Bushel 

Tua 

Electrics 

Tui 

Dividend 

Tuo 

Quotient 

Tu me 

Division of fractions 

Tuum 

Division 

T a 

For 

Tu 

Hogshead 

Tu 

Divisor 


165 


T a 

Pipe 

Tan 

Tun 

T via 

Remainder 

U 

Mills 


ENGLISH VOCABULARY. 


Addition 

Puum 

Addition of fractions 

Pume 

And 

De 

Arabic 

Ala 

Arithmetic 

Momo 

Arithmetic, the one 

Cho Momo 

Arithmetic, abstract 

Amomo 

Arithmetic, concrete 

A momo 

At 

a 

Billiineter 

Ae 

Bushel 

Tom 

Cardinal numbers 

Dili 

Center 

0 

Centimeter 

Ce 

Cents 

0 

Cipher 

0 

Circle of the earth 

Mei 

Circle of the zodiac 

Meim 

Co aid 

Koad 

Cube 

Ntm 



166 


Cube 8 

Nid a 

Cube, root of 

(D )a 

Day 

Ho 

Decimeter 

De 

Decameter 

Fe 

Difference 

Ruo 

Degree 

Lei 

Degree 

Eeim 

Denominator 

Dimo 

Digom 

Digom 

Dimes 

I 

Dividend 

Tim 

Division 

Tuum 

Division of fractions 

Tume 

Divisor 

Tu 

Dollars 

A 

Dram 

La 

Electrics 

Tua 

Electricity 

K#om 

Environment 

K a 

Equality 

Ke 

Equal to 

Ke 

Equivalent 

Ike 

Factors 

Sua 

Fingers, all the 

A<?n 

thumb, left 

On 

index, left 

Bon 

long, left 

Don 

ring, left 

Bon 


167 


Fingers 


little, left 

Gtfin 

little, right 

Kon 

companion, right 

hon 

long, right 

M<?n 

index, right 

N<?n 

thumb, right 

P<?n 

Figorn 

Figorn 

Foot 

Dei 

Foot, square 

Dea 

Foot, cube 

Dea 

For 

T a 

Fractions 

Dime 

Fractions, addition 

Pume 

Fractions, division 

Tume 

Fractions, multiplication 

Sume 

Fractions, subtraction 

Rume 

Gallon 

To 

Gigom 

Gigom 

Gill 

Ta 

Gom 

Gom 

‘Grain 

Goma 

Hectometer 

Ge 

Hogshead 

Tu 

Hours 

Hi 

Hundredweight 

Lu 

Immaterial 

Eom 

Imponderable 

On 

Inch 

Cei 


Interest 

Interrogation 

Is 

Kilometer 

League 

Magnets 

Magnetism 

Material 

Meter 

Metric system 
Mile 

Mile, square 

Millimeter 

Mills 

Minuend 

Minutes 

Minus 

Money 

Month 

Mule 

Muli 


Numerator 

Numbers 

Numbers, abstract 


Iteu 

Kal 

Ke 

He 

Kei 

Pua 

Chom 

Eon 

E. Cho E. 

Cho. E system 

Hei 

Hea 

Be 

U 

Rui 

He 

Ru 

Ou 

Ha 

Mule 

Muli 

Suutn 

Sume 

Sui 

Su 

Ie 

Dimi 

Dil 

Adil 


Multiplication 

Multiplication of fractions 

Multiplican 

Multiplier 

Myriameter 


169 


Numbers, concrete 

Adi\ 

Numbers, composite 

Bid 

Numbers, feminine 

Odil 

Numbers, masculine 

Bidil 

Numbers, mixed 

Dimu 

Numbers, ordinal 

Dile 

Numbers, prime 

Bib 

Objective case 

Od 

One 

Cho 

Ounce 

Le 

Ounce 

Lem 

Parentheses 

Bu 

1 st parentheses 

Bua 

2 d parentheses 

Bue 

3 d parentheses 

Bui 

4 th parentheses 

Buo 

Peck 

Tim 

Per cent 

Oteo 

Percentage 

Oteom 

Pint 

Te 

Pipe 

T a 

Plus 

Pu 

Pound 

Li 

Pound 

Lim 

Possessive case 

Ad 

Price, cost 

Lak 

Product 

Suo 


Quart 

170 

Ti 

Quarter 


Lo 

Quotient 


Tuo 

Ratio 


Rateo 

Remainder 


Tu a 

Rod 


Fei 

Rod, square 


Fea 

Rood 


Gei 

Rood, square 


Gea 

Seconds 


Ha 

Square 


Nua 

Subtraction 


Ruum 

Subtraction of fractions 

Rume 

Subtrahend 


Ru 

Sum 


Puo 

That, those 


Mu 

Them, of them 


Muad 

This, these 


Tu 

Time 


Him 

Ton 


L a 

Tun 


Tam 

Vacuum 


Ko 

Week 


Hu 

What 


Kal 

Yard 


Ei 

Year 


Ham 


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